optimal investment-reinsurance policy with stochastic...

Research ArticleOptimal Investment-Reinsurance Policy with StochasticInterest and Inflation Rates

Xin Zhang 1 and Xiaoxiao Zheng2

1School of Mathematics Southeast University Nanjing Jiangsu 211189 China2Department of Fixed Income Bohai Securities Co Ltd Tianjin 300381 China

Correspondence should be addressed to Xin Zhang xzhangseugmailcom

Received 13 September 2019 Revised 13 November 2019 Accepted 2 December 2019 Published 17 December 2019

Academic Editor Alberto Cavallo

Copyright copy 2019 Xin Zhang and Xiaoxiao Zheng )is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

)e aim of this paper is to study a classic problem in actuarial mathematics namely an optimal reinsurance-investment problemin the presence of stochastic interest and inflation rates )is is of relevance since insurers make investment and risk managementdecisions over a relatively long horizon where uncertainty about interest rate and inflation rate may have significant impacts onthese decisions We consider the situation where three investment opportunities namely a savings account a share and a bondare available to an insurer in a security market In the meantime the insurer transfers part of its insurance risk through acquiring aproportional reinsurance )e investment and reinsurance decisions are made so as to maximize an expected power utility onterminal wealth An explicit solution to the problem is derived for each of the two well-known stochastic interest rate modelsnamely the HondashLee model and the Vasicek model using standard techniques in stochastic optimal control theory Numericalexamples are presented to illustrate the impacts of the two different stochastic interest rate modeling assumptions on optimaldecision making of the insurer

1 Introduction

)e purpose of this paper is to investigate a classic problem inactuarial mathematics namely an optimal reinsurance-in-vestment problem in the presence of stochastic interest andinflation rates Investment and risk management decisions ofinsurance companies are by their nature long-term decisionmaking Over a relatively long period of time there could besignificant changes in interest rate and inflation rate andthese changes are usually difficult to be anticipated at the timewhile the decisions are made Consequently the in-corporation of randommovements of these rates in modelingdecision making of insurance companies may be of practicalinterest and relevance In this paper two stochastic interestrate models namely the HondashLee model and the Vasicekmodel are considered To incorporate a realistic feature ofinflation rate namely mean reversion we adopt a mean-reverting time-dependent OrnsteinndashUhlenbeck (OU) pro-cess A diffusion approximation is employed to model the

surplus process of an insurance company To incorporate theimpact of inflation on the surplus process which is relevantdue to the long-term nature of the decision making problemsof the insurance company the surplus process is adjusted tothe price index level reflecting inflation We consider thesituation where the insurance company can transfer its riskattributed to insurance liabilities to a reinsurance company bymeans of proportional reinsurance which is one of thepopular reinsurance treaties )e insurance company mayalso diversify its risk and make profits from investing in asecurity market It is supposed here that there are three in-vestment opportunities available to the insurance company ina security market namely a risky share a zero-coupon bondand a saving account )e aim of the insurance company is todetermine a combined optimal investment and reinsurancepolicy with a view to maximizing the expected power utilityon the companyrsquos terminal wealth Using the HamiltonndashJacobindashBellman (HJB) dynamic programming approach anexplicit solution to the optimal investment and reinsurance

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 5176172 14 pageshttpsdoiorg10115520195176172

problem of the insurance company is obtained in each of thetwo stochastic interest rate models namely the HondashLeemodel and the Vasicek model )e verification theorems forthe HJB solutions to the optimal investment and reinsuranceproblem for the two stochastic interest rate models areprovided Numerical results based on hypothetical parametervalues are presented to illustrate the impacts of the twodifferent stochastic interest rate modeling assumptions onoptimal decision making of the insurer In what follows abrief review on some relevant literature and the motivation ofthis study are provided

)e optimal investment-reinsurance problem is one ofthe classic research topics in actuarial and insurancemathematics It aims to provide a theoretical basis for ra-tional decision making of insurers on investments in a se-curity market and transferring insurance risks toreinsurance companies )e rational decisions of the in-surers are made by optimizing certain objective criteria suchas utility function )is is related to the utility maximizationproblem in neoclassical economics as well as financialeconomics )e solution of the optimization problem is byits nature normative though results of positive nature mayoccasionally obtain In financial economics a ldquoformalrdquo studyof an optimal investment problem may be tracked back tothe pioneering work by Markowitz [1] where the scientificfoundation of the problem was enacted by making use ofmathematics Specifically the problem was formulatedmathematically as an optimization problem in a static one-period model where only the expected return of a portfolioand the risk of the portfolio measured by the variance of itsreturn were relevant More realistic situations such ascontinuous-time situations were considered in the secondstage of developments pioneered by the seminal articles byMerton [2 3] In particular the continuous-time optimalasset allocation model by Merton [2 3] brought the his-torical developments of the field in a new era and stimulatedthe development of continuous-time finance (see for ex-ample Siu [4] for related discussion) Since the seminalworks by Merton [2 3] there have been many significantdevelopments in continuous-time optimal asset allocationmodels and a large volume of literature on the field It doesnot seem to be easy to list all of them here but some of themare mentioned without slighting the other literature Onemay refer to the monographs by Merton [5] Ralf [6]Karatzas and Shreve [7] Elliott and Kopp [8] and therelevant literature therein for more detailed exposition onthe developments of the literature in the field Davis andNorman [9] investigated an optimal consumption and in-vestment decision for an investor who invests in a simplifiedsecurity market with a bank account and a stock Fur-thermore the authors took fixed percentage transactioncosts into consideration A life-cycle model of consumptionand portfolio was considered by Cocco et al [10] Kraft [11]studied an optimal portfolio selection problem with Hes-tonrsquos stochastic volatility

Long-term strategic asset allocation is one of the im-portant areas in optimal asset allocation It sheds light onoptimal investment decision making in financial planninginsurance and pension funds Intuitively factors such as

inflation interest rate investment cycle and economic cyclewhich seem to be relevant to long-term investment decisionmaking may be taken into account when discussing long-term strategic asset allocation )e monograph by Campbelland Viceira [12] provides an excellent account of long-termstrategic asset allocation where the problem was rigorouslytreated in discrete time modeling frameworks)ere is also aconsiderable amount of finance and insurance literature onlong-term strategic asset allocation Again some literaturestudies particularly those relating to incorporation of in-flation and interest rate are mentioned here Inflation has asignificant impact on long-term investment decision mak-ing Indeed inflation was used as the main driver of in-vestment return series in long run in one of the majormodels in actuarial science namely the Wilkie stochasticinvestment model developed by Wilkie [13] A key featureof inflation namely mean reversion was described by anautoregressive time series model (in discrete time) in theWilkie stochastic investmentmodel (see for example Siu [4]for related discussion) Pearson and Sun [14] described theinflation risk by a mean-reverting square-root process whilein Munk et al [15] and Brennan and Xia [16] the inflationprocess is modeled by an OrnsteinndashUhlenbeck process Siu[4] discussed a long-term strategic asset allocation problemin continuous time by incorporating inflation and regimeswitching Korn et al [17] and Siu [18] considered an op-timal asset allocation of pension funds in the presence ofboth the inflation and regime switching risks using amartingale approach and a backward stochastic differentialequation approach respectively See also the relevant lit-erature in the papers by Siu [4] Korn et al [17] and Siu [18]for some works on optimal asset allocation in the presence ofinflation Apart from inflation risk the interest risk is an-other important factor that we have to face Korn and Kraft[19] studied the portfolio problems with stochastic interestrates In Li and Wu [20] stochastic interest rate is given bythe CoxndashIngersollndashRoss (CIR) model and the volatility of thestock is also a CIR process Kraft [21] considered all commonshort rate models and stochastic discount Shen and Siu [22]investigated an optimal asset allocation problem in thepresence of both stochastic interest rate and regimeswitching effect where a regime-switching version of theVasicek model was used to describe stochastic interest rate

)ere has been a large amount of work on optimalinvestment of an insurance company Some of the literaturestudies are for example Wang et al [23] Zeng [24] Luoet al [25] Elliott and Siu [26] Liang et al [27] Elliott and Siu[28] and Siu [29] For an insurance company it is natural toadopt a prudent investment strategy that is they should notonly diversify risk by investing their money into differentassets but also should avoid the risk brought from thecompanyrsquos relatively long investment cycle Over a longperiod of time accumulated inflation can lead to hugeshrinkage of the wealth What is more relevant here is acombined optimal investment and reinsurance problem ofan insurer Some of the literature is also mentioned here Baiand Zhang [30] considered an optimal investment-re-insurance problem for an insurer under the mean-variancecriterion Zhang and Siu [31] incorporated model

2 Mathematical Problems in Engineering

uncertainty or ambiguity in an optimal investment-re-insurance problem by adopting a robust approach Zhangand Siu [32] discussed an optimal proportional reinsuranceand investment problem using the criterion of maximizingutility function on terminal wealth Zhang et al [33] con-sidered a Bayesian approach to incorporate parameter un-certainty in an optimal reinsurance and investment problemunder a diffusionmodel for the surplus process of an insurerLiu et al [34] considered an optimal insurance-reinsuranceproblem in the presence of dynamic risk constraint andregime switching

As we know both the stochastic interest rates and in-flation cannot be ignored for decision makers when dealingwith long-term decision problems )erefore the in-corporation of both stochastic interest rate and inflationmakes our problems more practicable for the long-termdecisionmaking and this is the first contribution of our paper)e second contribution of our paper is that we establish averification theorem for the solution using an approach whichis not the same as the usual approach adopted for proving averification theorem In fact unlike previous literaturestudies the usual standard technical conditions required bythe verification theorem for the solution of the optimal in-vestment-reinsurance problem such as the Lipschitz condi-tion and the linear growth condition are not satisfied by themodel considered which makes the proof of the verificationtheorem to be a challenging thing

)e rest of this paper is organized as follows Section 2presents the model dynamics and assumptions in theeconomy finance and insurance markets )e optimizationproblem of the insurer is formulated in Section 3 Fur-thermore the closed-form solutions to the problem underthe two stochastic interest rate models are derived by in-voking the use of the HJB dynamic programming approachIn Section 4 the verification theorem for the solutions of theproblem under the two models is established Numericalcomparison and analysis are presented in Section 5)e finalsection gives some concluding remarks )e proofs of someresults are placed in an Appendix A

2 Model Dynamics and Assumptions

To begin with as usual we describe uncertainties in thecontinuous-time economy finance and insurance marketsas well as their information flows by a complete filteredprobability space (ΩF F p) where F Ft1113864 1113865tisin[0T] is rightcontinuous p is the complete filtration and p is a real-world probability For simplicity it is supposed that thefiltration is ldquosufficiently largerdquo so that all of the processes tobe defined below are F-adapted

21 Price Index )e key economic variable we are intendingto model is inflation )is is relevant to the optimal re-insurance and investment problem of an insurer since thedecision making horizon of an insurer is usually long say adecade or more and inflation is a key economic factor whichcould lead to diminution of financial wealth of the insurerInflation can be measured by an inflation rate which may be

proxied by the annualized percentage change in a price indexExamples of a price index are consumer price indices andretail price indices Incorporating inflation in studying op-timal asset allocation problem has been considered in theliterature Some examples are Munk et al [15] Pearson andSun [14] Brennan and Xia [16] Douglas [35] George [36]and Fama and Gibbons [37] It is usually assumed that a priceindex which is used to proxy an inflation rate is modeled by astochastic process Here as in Munk et al [15] and Yao et al[38] it is supposed that the evolution of the nominal priceindex of a consumption good in an economy over time isgoverned by the following geometric Brownian motion withstochastic drift

dΠ(t) Π(t) I(t)dt + σ0(t)dW0(t)1113858 1113859

for all t isin [0 T] Π(0) Π0(1)

where W0(t)1113864 1113865tisin[0T] is a standard Brownian motion σ0(t) isthe volatility of the price index and the stochastic drift I(t)

represents the instantaneous expected inflation rate whoseevolution over time is assumed to be governed by the fol-lowing time-dependent OrnsteinndashUhlenbeck (OU) process

dI(t) β(t)[α(t) minus I(t)]dt + σ0(t)dW0(t) (2)

Here α(t) describes the long-run mean of the expectedinflation rate β(t) represents the degree of mean-reversionand σ0(t) is the volatility of the expected inflation rateConsequently the mean-reverting property of the expectedinflation rate is incorporated in themodel considered here Tosimplify our discussion it is assumed that σ0(t) α(t) β(t)and σ0(t) are deterministic and continuous functions of timet isin [0 T] However we adopt here a time-dependent OUprocess rather than the one in Munk et al [15] and Yao et al[38]

22 Financial Market Assume that the financial marketconsidered here consists of three assets one savings accountone stock and one zero-coupon bond with maturity T1 gtT

Let B(t) tisin[0T] denote the price process of the savingsaccount Assume that the evolution of B(t) tisin[0T] over timeis determined by

dB(t) r(t)B(t)dt B(0) 1 (3)

Here r(t) is the short-term interest rate or the spot ratewhose evolution over time is governed by the followingstochastic differential equation

dr(t) a(t)dt + bdW1(t) t isin 0 T11113858 1113859 r(0) r0 (4)

where b is a positive constant and W1(t)1113864 1113865tisin[0T] is a standardBrownian motion Following some explicit examples inKorn and Kraft [19] the HondashLee model and the Vasicekmodel are considered here where a(t) is respectively givenby a(t) 1113957a(t) + bξ(t) and a(t) θ(t) minus 1113954br(t) + bξ(t) Herethe risk premium ξ(t) and parameter θ(t) are assumed to bedeterministic and continuous functions of time t isin [0 T]

Let P(t T1) denote the price process of a zero-couponbond with maturity T1 gtT )en from Korn and Kraft [19]P(t T1) satisfies the following stochastic differentialequation

Mathematical Problems in Engineering 3

dP t T1( 1113857 P t T1( 1113857 r(t) + ξ(t)σ1(t)1113858 1113859dt + σ1(t)dW1(t)1113864 1113865

P 0 T1( 1113857 P0 gt 0

(5)

where r(t) is given by (4) In what follows we shall writeP(t) for P(t T1) to ease the notation As shown in Korn andKraft [19] the volatilities of the zero-coupon bond under theHondashLee model and Vasicek model are given byσ(t) minus b(T1 minus t) and σ(t) (b1113954b)[exp minus 1113954b(T1 minus t)1113966 1113967 minus 1]respectively

Furthermore it is supposed that the stock price processS(t) tisin[0T] evolves over time according to the followinggeometric Brownian motion

dS(t) S(t) μ(t)dt + σ2(t)dW2(t)1113858 1113859 S(0) S0 (6)

where W2(t)1113864 1113865tisin[0T] is a standard Brownian motion andσ2(t) is the volatility of the stock at time t where it is as-sumed that σ2(t) is a deterministic and continuous functionof time t As in Korn and Kraft [19] we decompose theappreciation rate μ(t) of the stock into the sum of a liquiditypremium (LP) and a risk premium (RP)

μ(t) r(t)1113980radic11139791113978radic1113981r(t)

+ μ(t) minus r(t)1113980radicradicradicradic11139791113978radicradicradicradic1113981

RP

(7)

Let 1113957λ(t) denote the risk premium of the stock ie1113957λ(t) ≔ μ(t) minus r(t) Consequently the price process of thestock can be rewritten as

dS(t) S(t) [r(t) + 1113957λ(t)]dt + σ2(t)dW2(t)1113966 1113967 (8)

23 Surplus Process Let 1113957R(t) be the surplus of an insurancecompany at time t isin [0 T] Since optimal long-term in-vestment and reinsurance decision making of the insurancecompany are considered it is relevant to incorporate theimpact of inflation on the surplus of the insurance companyConsequently it is supposed here that the increment of thesurplus of the company depends on the price index de-scribed in Section 21 Specifically without loss of generalityit is assumed that the surplus process of the company1113957R(t)1113864 1113865tisin[0T] evolves over time according to a diffusion ap-proximation model with the impact of the price index beingincorporated as follows

d1113957R(t) Π(t)c(t)dt + Π(t)σ3(t)dW3(t) (9)

where W3(t)1113864 1113865tisin[0T] is a standard Brownian motion and c(t)

and σ3(t) are deterministic and continuous function of timet isin [0 T] with c(t)gt 0 Note that c(t) and σ3(t) may berespectively interpreted as the real premium rate of theinsurer and the risk attributed to uncertainty about futureinsurance liabilities See for example Jan [39] Zeng and Li[40] Hoslashjgaard and Taksar [41] and Michael and Markussen[42] for more details about diffusion approximation modelsto surplus processes of insurance companies

Remark 1 Note that d1113957R(t) denotes the instantaneous in-crement that incorporates the impact of price indexEquation (9) indicates that the instantaneous increment

changes in the surplus depends on the price index Π(t) Inother words the real instantaneous increment change isgiven by c(t)dt + σ(t)dW3(t)

Let u(t) be the proportional reinsurance retention leveladopted by the insurance company at time t whereu(t) isin [0infin) Assume for simplicity that the safety load-ings of the insurance company and the reinsurance companyare the same Consequently the surplus process of insurancecompany R(t) tisin[0T] after acquiring the proportional re-insurance is given by

dR(t) Π(t)u(t)c(t)dt + Π(t)u(t)σ3(t)dW3(t) (10)

Lastly we assume that the stochastic interest rate thebond price the price index and the expected inflation ratecould be correlated Without loss of generality we supposeCov(W1(t) W0(t)) ρt ρ isin (minus 1 1) Also assume thatW2(t)1113864 1113865tisin[0T] and W3(t)1113864 1113865tisin[0T] are independent Brownianmotions which are independent of W1(t)1113864 1113865tisin[0T] andW0(t)1113864 1113865tisin[0T] If we suppose the interest rate or the priceindex is correlated with the stock there will be an additionalmixed partial derivative term in HJB equation but it will notaffect the method used in the remainder of the paper

24 Wealth Process Suppose that the insurer is allowed tocontinuously purchase proportional reinsurance and investsall of his (or her) wealth in the financial market over the time[0 T] with TltT1 )is assumption is an idealisation in thecontinuous-time modeling set up here In practice an in-surer may only be able to acquire reinsurance treaties indiscrete time periods and there are frictional or transactioncosts when the insurer invests in the security market

Let π1(t) and π2(t) be the proportions of the total wealthinvested in the bond and stock at time t respectivelySuppose u(t) denotes the proportional reinsurance re-tention level at time t Accordingly 1 minus π1(t) minus π2(t) is theproportion of the total wealth invested in the saving accountIf we denote by 1113957X(t) the wealth of the insurer at time t afteradopting the reinsurance and investment then we have

d 1113957X(t) π1(t) 1113957X(t)dP(t)

P(t)+ π2(t) 1113957X(t)

dS(t)

S(t)

+ 1 minus π1(t) minus π2(t)( 1113857 1113957X(t)dB(t)

B(t)+ Π(t)u(t)c(t)dt

+ Π(t)u(t)σ3(t)dW3(t)

1113957X(t)1113882 r(t) + π1(t)ξ(t)σ1(t) + π2(t)1113957λ(t)1113960 1113961dt

+ π1(t)σ1(t)dW1(t) + π2(t)σ2(t)dW2(t)1113883

+ Π(t)u(t)c(t)dt + Π(t)u(t)σ3(t)dW3(t)

(11)

In Section 23 the nominal price index of the con-sumption good in the economy at time t is denoted by Π(t)


)e real price of an asset in the economy over a long time isdetermined by deflating its nominal value with the priceindex Π(t) )e real wealth of the insurer which adjusts forthe impact of inflation is given by X(t) 1113957X(t)Π(t) )enapplying the Ito formula (see for example Karatzas andShreve [43]) gives

dX(t) X(t) r(t) + σ20(t) minus I(t) + ξ(t)σ1(t)(11139601113966

minus ρσ1(t)σ0(t)1113857π1(t) + 1113957λ(t)π2(t)1113961 + u(t)c(t)1113967dt

+ u(t)σ3(t)dW3(t) + X(t)π1(t)σ1(t)dW1(t)

+ X(t)π2(t)σ2(t)dW2(t) minus X(t)σ0(t)dW0(t)

(12)

and the initial value X(0) ( 1113957X(0)Π(0)) X0In what follows we take η(t) ξ(t) minus ρσ0(t) With a

slight abuse of the notation we use λ(t) to denote 1113957λ(t)σ2(t)Consequently we can rewrite the wealth process as

dX(t) 1113882X(t)1113876r(t) + σ20(t) minus I(t) + σ1(t)η(t)π1(t)

+ λ(t)σ2(t)π2(t)1113877 + u(t)c(t)1113883dt



(13)

with X(0) X0Next we give the definition of an admissible control

Definition 1 A strategy 1113957π(t) (π1(t) π2(t) u(t)) is said tobe admissible if (1) π1(t) π2(t) and u(t) are progressively

measurable processes (2) π1(t) and π2(t) are bounded (3)u(t)ge 0 (4) 1113957π(t) leads to a positive wealth process Wedenote Θ by the set of all admissible controls

3 Maximizing the Expected Power Utility

In this section we first present the optimal investment andreinsurance problem of an insurer as an utility maximizationproblem where the insurer aims to select an investment-reinsurance mix to maximize the expected power utility onterminal wealth )en using the standard dynamic pro-gramming approach (see for example Wendell [44] andFleming and Soner [45]) we derive the HamiltonndashJacobindashBellman (HJB) equation governing the value function of theutility maximization problem

Suppose that the insurerrsquos preference is described by anon-log hyperbolic absolute risk aversion (HARA) utilityfunction U(x) (1p)xp 0ltplt 1 xgt 0 and that the in-surer wishes to maximize the expected utility of terminalwealth at time T For any strategy 1113957π isin Θ the expected powerutility of the insurer at time T denoted by Vπ1113957(t x r I) isgiven by

Vπ1113957(t x r I) E1p

Xπ1113957T1113874 1113875

p1113868111386811138681113868111386811138681113868Xt x rt r It I1113896 1113897 (14)

where the dynamics of Xπ1113957t1113882 1113883

tisin[0T] rt1113864 1113865tisin[0T] and It1113864 1113865tisin[0T]

are governed by the following three stochastic differentialequations

dX(t) X(t) r(t) + σ20(t) minus I(t) + σ1(t)η(t)π1(t) + λ(t)σ2(t)π2(t)1113858 1113859 + u(t)c(t)1113864 1113865dt

+ u(t)σ3(t)dW3(t) + X(t)π1(t)σ1(t)dW1(t) + X(t)π2(t)σ2(t)dW2(t) minus X(t)σ0(t)dW0(t)

dr(t) a(t)dt + bdW1(t)

dI(t) β(t)[α(t) minus I(t)]dt + σ0(t)dW0(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(15)

Our goal is to find the value function

V(t x r I) supπ1113957isinΘ

Vπ1113957(t x r I) (16)

and the optimal strategy 1113957πlowast (πlowast1 πlowast2 ulowast) such that

V(t x r I) V~πlowast(t x r I) (17)

Using standard arguments in dynamic programmingthe value function satisfies the following HJB equation withthe corresponding terminal condition as follows


0 Vt + Vx r minus I + σ20(t)1113960 1113961x +12Vxxσ

20(t)x

2+ a(t)Vr

+12b2Vrr + β(t)[α(t) minus I]VI

+12σ20(t)VII minus ρσ0(t)bxVxr minus σ0(t)σ0(t)xVxI

+ bσ0(t)ρVrI + supπ1isinR

111388212Vxxσ

21(t)x

2π21 + 1113876Vxσ1(t)η(t)x

minus Vxxρσ1(t)σ0(t)x2

+ Vxrbσ1(t)x

+ VxIσ0(t)σ1(t)ρx1113877π11113883

+ supπ2isinR

12Vxxσ

22(t)x

2π22 + Vxλ(t)σ2(t)xπ21113882 1113883

+ supuisin[0infin)

12Vxxσ

23(t)u

2+ c(t)Vxu1113882 1113883

(18)

V(T x I r) 1p

xp (19)

In what follows we first present a solution to HJBequation (18) with terminal condition (19) Suppose thatHJB equation (18) with terminal condition (19) has aclassical solution G satisfying the conditions that Gx gt 0 andthat Gxx lt 0 Furthermore we assume that the solutionG hasthe following form

G(t x r I) g(t r I)xp

p (20)

where g(t r I) is a function with the terminal conditiong(T r I) 1 for all I and r and it will be determined in thesequel

Differentiating the left hand side of equation (18) withrespect to π1 π2 and u and setting the derivatives equal to 0yield the following candidate optimal controls

πlowast1(t) minusη(t)

σ1(t)

Gx

xGxx

minusσ0(t)ρσ1(t)

GIx

xGxx

minusb

σ1(t)

Gxr

xGxx

+σ0(t)ρσ1(t)

(21)

πlowast2(t) minusGx

xGxx

λ(t)

σ2(t) (22)

ulowast(t) minus

Gx

Gxx

c(t)

σ23(t) (23)

Note that for the function G we have

Gt gt

xp

p

Gx gxpminus 1

Gxx g(p minus 1)xpminus 2

Gr gr

xp

p

Grr grr

xp

p

GI gI

xp

p

GII gII

xp

p

Gxr grxpminus 1

GxI gIxpminus 1

GrI grI

xp

p

(24)

Consequently substituting the above equations into (18)leads to

0 gt

p+ g r + σ20 minus I1113872 1113873 +

12σ20g(p minus 1) + agr

1p

+12b2grr

p

+gI

pβ(α minus I) +

12σ20

gII

pminus σ0σ0gI minus grρσ0b +

gIr

pbρσ0

minusg

p minus 1η2

2minus12

g(p minus 1)σ20ρ2

minusg2

I

g(p minus 1)

σ20ρ2

2

minusg2

r

g(p minus 1)

b2

2+ gησ0ρ minus

gI

p minus 1ηρσ0 minus

gr

p minus 1bη

+ grbσ0ρ + gIσ0σ0ρ2

minusgrgI

g(p minus 1)σ0ρb minus

g

p minus 1λ2

2

minusg

p minus 1c2

2σ23

(25)

Consider the following trial solution

g(t r I) f(t)ek(t)r+z(t)I

(26)

where f(t) k(T) and z(t) are the functions of time t withtheir respective terminal values f(T) 1 k(T) 0 andz(T) 0 Let Δ ≔ k(t)r + z(t)I )en


gt fprimeeΔ + feΔ

kprimer + zprimeI( 1113857

gr fkeΔ

gI fzeΔ

gIr fkzeΔ

grr fk2eΔ

gII fz2eΔ

(27)

Substituting them into (25) yields

0 kprimep

+ 11113890 1113891fr +zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ 1113882σ20 +12

(p minus 1)σ20 +b2k2

2p+αβp

z +σ20z

2

2p

minus σ0σ0z minus kρσ0b +bρσ0

pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2

+ σ0ρη minusηρσ0p minus 1

z

minusηb

p minus 1k + σ0σ0ρ

2z + σ0bρk minus

σ0ρb

p minus 1kz minus

λ2

2(p minus 1)

minusc2

2σ23(p minus 1)1113883f +

ak

pf

(28)

Due to the different forms of a(t) for the HondashLee modeland Vasicek model we solve the above differential equationseparately

31 HondashLee Model Let

h(t) ≔ σ20 +12

(p minus 1)σ20 +(1113957a + bξ)k

p+

b2k2

2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz

minusλ2

2(p minus 1)minus

c2

2σ23(p minus 1)

(29)

Suppose that k(t) z(t) andf(t) are the solutions to thefollowing ordinary differential equations

kprime(t) minus p

k(T) 0

⎧⎨

⎩ (30)

zprime(t) minus β(t)z(t) minus p 0

z(T) 01113896 (31)

fprime(t) + ph(t)f(t) 0

f(T) 11113896 (32)

Note that for the HondashLee model a(t) ≔ 1113957a(t) + bξ(t))erefore we can rewrite (28) as

0 kprimep

+ 11113890 1113891fr +zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ h(t)f (33)

It can be seen that the functions k(t) z(t) andf(t)

determined by (30)ndash(32) satisfy equation (33) Using stan-dard theory of ordinary differential equations explicit ex-pressions for k(t) z(t) andf(t) are obtained as follows

k(t) p(T minus t)

z(t) minus pe1113938

t

0β(s)ds

1113946

T

t

eminus 1113938

s

0β(v)dvds

f(t) eminus p 1113938

T

th(s)ds

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)

Combining (20) (26) and (34) an explicit solution toHJB (18) with terminal condition (19) is obtained asfollows

G(t x r I) 1pexp minus p 1113946

T

th(s)ds + k(t)r + z(t)I1113896 1113897x

p

(35)

where k(t) and z(t) are given by (34) Now from equations(21)ndash(23) the following expressions for the candidate op-timal controls are obtained


σ1(t)

1p minus 1

minusb

σ1(t)

p

p minus 1(T minus t) +

ρσ0(t)

σ1(t)

minusρσ0(t)

σ1(t)

p

p minus 1e1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds

πlowast2(t) minusλ(t)

σ2(t)

1p minus 1

ulowast(t) minus

c(t)

σ23(t)

1p minus 1

x

(36)


32 Vasicek Model Let

1113957h(t) ≔ σ20 +12

(p minus 1)σ20 +(θ + bξ)k

p+

b2k2

2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz minus

λ2

2(p minus 1)

minusc2

2σ23(p minus 1)

(37)

Suppose that 1113957k(t) 1113957z(t) and 1113957f(t) are the solutions to thefollowing ordinary differential equations

1113957kprime(t) minus 1113954bk(t) + p 0

1113957k(T) 0

⎧⎨

⎩ (38)

1113957zprime(t) minus β(t)1113957z(t) minus p 0

1113957z(T) 01113896 (39)

1113957fprime(t) + p1113957h(t)1113957f(t) 0

1113957f(T) 11113896 (40)

Note that for the Vasicek modela(t) θ(t) minus 1113954br(t) + bξ(t) and so (28) is equivalent to

0 kprimep

minus1113954b

pk + 11113890 1113891fr +

zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ 1113957hf (41)

It can be seen that the functions 1113957k(t) 1113957z(t) and 1113957f(t)

determined by (38)ndash(40) satisfy equation (41) Again usingthe standard theory of ordinary differential equations thefollowing explicit expressions for 1113957k(t) 1113957z(t) and 1113957f(t) underthe Vasicek model are obtained

1113957k(t) p

1113954b11138761 minus exp 1113954b(t minus T)1113966 11139671113877

1113957z(t) minus pe1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds

1113957f(t) eminus p 1113938

T

t

~h(s)ds

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)

Combining (20) (26) and (42) an explicit solution forHJB equation (18) with terminal condition (19) under theVasicek model is obtained as follows


T

t

1113957h(s)ds + 1113957k(t)r + 1113957z(t)I1113896 1113897xp

(43)

where 1113957k(t) and 1113957z(t) are given by (42) Now from equations(21)ndash(23) the following expressions for the candidate op-timal controls under the Vasicek model are obtained


σ1(t)

1p minus 1

minus1

σ1(t)

p

p minus 11 minus e

b(tminus T)1113960 1113961

minusρσ0(t)

σ1(t)

p

p minus 1e1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds +

ρσ0(t)

σ1(t)


σ2(t)

1p minus 1

ulowast(t) minusc(t)

σ23(t)

1p minus 1

x

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)

4 Verification Theorem

A verification theorem for the solution to HJB (18) derived inthe previous section is presented for the two stochasticinterest rate models )e verification theorem states that thesolution of HJB (18) is the value of the optimal controlproblem of the insurer and the candidate optimal control isan optimal control

Due to the presence of r(t)X(t) and I(t)X(t) in thewealth process of (13) the usual verification theorem whichrequires Lipschitz and linear growth conditions is not ap-plicable in our situation Inspired by the methods used byKraft [11] Li and Wu [20] and Kraft [21] we need to someuniform integrability of G(τn Xlowastτn

I(τn) r(τn))1113966 1113967nisinN where

Xlowastt1113864 1113865 is the optimal wealth process and τn1113864 1113865nisinN is a sequenceof stopping times which is bounded above by T

Lemma 1 Let Xlowastt1113864 1113865 denote the respective optimal wealthprocess Ben

(1) HondashLee model suppose G and1113957πlowast(t) (πlowast1(t) πlowast2(t) ulowast(t)) are given by (35) and(36) respectively Ben the sequenceG(τn Xlowastτn

I(τn) r(τn))1113966 1113967nisinN is uniformly integrable

for all sequences of stopping times τn1113864 1113865nisinN with boundT

(2) Vasicek model suppose G and1113957πlowast(t) (πlowast1(t) πlowast2(t) ulowast(t)) are given by (43) and(44) respectively G(τn Xlowastτn

I(τn) r(τn))1113966 1113967nisinN is

uniformly integrable for all sequences of stoppingtimes τn1113864 1113865nisinN with bound T


)emain idea of proving the above lemma is from Kraft[11] Li andWu [20] and Kraft [21] but there still exist somedetails different from these references)erefore we providethe proof in the appendix for the sake of completeness Basedon the uniform integrability property we can prove theverification theorem

Theorem 1 (verification theorem for two models)

(1) HondashLee model Suppose G(t x r I) is defined by (35)then for any 1113957π(t) (π1(t) π2(t) u(t)) isin Θ we have

EtxrI

1p

Xπ1113957T1113874 1113875

p

1113890 1113891leG(t x r I) (45)

Furthermore the candidate optimal control 1113957πlowast de-fined by (36) is indeed the optimal control and

V(t x r I) EtxrI

1p

Xπ1113957lowastT1113874 1113875

p

1113890 1113891 G(t x r I) (46)

(2) Vasicek model Suppose G(t x r I) is defined by (43)then for any 1113957π(t) (π1(t) π2(t) u(t)) isin Θ we have

EtxrI

1p

Xπ1113957T1113874 1113875

p

1113890 1113891leG(t x r I) (47)


V(t x r I) EtxrI

1p

Xπ1113957lowastT1113874 1113875

p

1113890 1113891 G(t x r I) (48)

Proof By Lemma 1 we know that G(t x r I) and 1113957πlowast satisfythe so-called ldquoproperty Urdquo of Definition 42 given by Kraft[21] )erefore applying the method used in)eorem 41 byKraft [21] leads to the desired results of the theorem

5 Numerical Analysis

In this section we shall present numerical analysis for theoptimal strategies under the two stochastic interest ratemodels For simplicity we assume the parameters areconstant over time interval t isin [0 T] Furthermore we takesome hypothetical values T 80 T1 120 η 00606b 005 ρ minus 006 and β 002 σ0 001 and σ0 0026

)e parameter p 0ltplt 1 in the utility functionrepresents the degree of risk aversion For example themore risk averse the investor is the larger the parameteris Figures 1 and 2 depict the changes in the proportion ofthe wealth invested in the bond for insurers having dif-ferent attitudes toward risk say different values of theparameter of p From Figures 1 and 2 it can be seen thatunder each of the two stochastic interest models say theHondashLee model and the Vasicek model the insurergradually increases the optimal proportion invested in

bond as time passes by On the other hand the two figuresreveal that an investor who is more risk averse will investless amount of money in the bond than the one who is lessrisk averse

Figure 3 provides a comparison for the optimal policiesof the insurer under the HondashLee model and the Vasicekmodel In this case we choose p 05 which represents acertain degree of risk aversion of the insurer From thisfigure it can be seen that if the HondashLee model is used todescribe the stochastic interest rate more money is investedin the bond than when the stochastic interest rate is modeledby the Vasicek model )e main feature that is described bythe Vasicek model is that the interest rate will revert to acertain long-run mean level Consequently the numericalresults reveal that the mean-reverting property of the

p = 04p = 05p = 06

0 10 20 30 40 50 60 70 80minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Figure 2 Case of the Vasicek model

0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

p = 04p = 05p = 06

Figure 1 Case of the HondashLee model


stochastic interest rate may lead to a reduction in the optimalamount of money invested in the bond by the insurer

6 Conclusion

)e contribution of the present paper rests on the in-corporation of both stochastic interest rate and inflation onlong-term decision making of an insurance companySpecifically the HondashLee model and the Vasicek model areemployed to describe the stochastic interest rate A chal-lenging aspect of the problem is that certain standardtechnical conditions such as the Lipschitz condition and thelinear growth condition which are required in a verificationtheorem for the solution of the optimal investment-re-insurance problem are not satisfied by the model consid-ered Here we establish a verification theorem for thesolution using an approach which is not the same as theusual approach adopted for proving a verification theoremFurthermore a closed-form solution to the optimal re-insurance-investment problem is obtained for each of thetwo stochastic interest rate models )is may make theimplementation of the optimal solutions easier To shed lighton understanding some implications of the optimal solu-tions numerical analysis and comparison for the optimalsolutions are provided to illustrate the impacts of the twodifferent stochastic interest rate modeling assumptions saythe HondashLee model and the Vasicek model on the optimaldecision making of the insurer It is found that the presenceof mean-reverting effect in the stochastic interest rate de-scribed by the Vasicek model reduces the optimal amount ofmoney the insurer should invest in the bond Inspired by Buiet al [46] and Wang et al [47] we will study our problemunder the game frameworks or more complex hybrid sto-chastic systems in the future

Appendix

A The Proof of Uniformly Integrability

Proof of Lemma 1 Note that the candidate optimal controlsulowast(t) in the HondashLee model are the same as that in theVasicek model (see (36) and (44)))us under the candidateoptimal control 1113957πlowast(t) (πlowast1(t) πlowast2(t) ulowast(t)) we can re-write the wealth process (13) as

dXlowastt X

lowastt 11138821113876r(t) + πlowast1(t)η(t)σ1(t) + πlowast2(t)σ2(t)λ(t)

minus I(t) + σ20(t) +c2(t)

σ23(t)

11 minus p

1113877dt +c(t)

σ3(t)

11 minus p

dW3(t)

+ σ1(t)πlowast1(t)dW1(t) + σ2(t)πlowast2(t)dW2(t)

minus σ0(t)dW0(t)1113883

(A1)

From the standard stochastic differential equation the-ory we have the following explicit expression of the wealthprocess

Xlowastt D1(t) middot exp1113882 1113946

t

0r(s)ds minus 1113946

t

0I(s)ds

+1

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + 1113946

t

0σ1(s)πlowast1(s)dW1(s)

+ 1113946t

0σ2(s)πlowast2(s)dW2(s) minus 1113946

t

0σ0(s)dW0(s)1113883

(A2)

where

D1(t) X0 exp1113946t

01113876πlowast1(s)η(s)σ1(s) + πlowast2(s)σ2(s)λ(s)

+ σ20(s) +c2(s)

(1 minus p)σ23(s)minus

c2(s)

2(1 minus p)2σ23(s)

minusσ21(s) πlowast1(s)( 1113857

2

2minusσ22(s) πlowast1(s)( 1113857

2

2minusσ20(s)

2

+ σ1(s)σ0(s)πlowast1(s)ρ1113877ds

(A3)

and X0 is the initial value of the wealth process

A1 HondashLee Model To prove the uniformly integrability ofG(τn Xlowastτn

I(τn) r(τn))1113966 1113967nisinN we only need to show that for

any fixed qgt 1

0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

HondashLee modelVasicek model

Figure 3 Comparison of the HondashLee model and Vasicek model onthe position of bond


supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A4)Combining equations (35) and (A2) we have for every

fixed qgt 1

G t Xlowastt r(t) I(t)( 1113857

11138681113868111386811138681113868111386811138681113868q

1

pqexp minus pq 1113946

T

th(s)ds + qk(t)r(t) + qz(t)I(t)1113896 1113897 X

lowastt( 1113857

pq

D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qk(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897

times exp qz(t)I(t) minus qp 1113946t

0I(s)ds minus qp 1113946

t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A5)

where we set

D2(t) 1pexp minus p 1113946

T

th(s)ds1113896 1113897D

p1(t) (A6)

Note that k(t) p(T minus t) r(t)t 1113938t

0 sdr(s) + 1113938t

0 r(s)dsand stochastic differential equation (4) satisfied by r(t) wecan easily obtain

exp qk(t)r(t) + qp 1113946t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897

exp qpTr0 + qp 1113946t

0a(s)(T minus s)ds1113896

+ qp 1113946t

0σ1(t)πlowast1(s) + b(T minus s)1113858 1113859dW1(s)1113897

(A7)

Applying the product differential rule to z(t)I(t) yields

z(t)I(t) z(0)I0 + 1113946t

0I(s)zprime(s)ds + 1113946

t

0z(s)dI(s)

z(0)I0 + 1113946t


t

0z(s) β(s)[α(s)1113864

minus I(s)]ds + σ0(s)dW0(s)1113865

z(0)I0 + 1113946t

0pI(s)ds + 1113946

t

0z(s)β(s)α(s)ds

+ 1113946t

0σ0(s)z(s)dW0(s)

(A8)

where we use equation (31) in the last equality)erefore wehave

exp qz(t)I(t) minus qp 1113946t


t

0σ0(s)dW0(s)1113896 1113897

exp1113882qz(0)I0 + q 1113946t

0β(s)α(s)z(s)ds

+ q 1113946t

0σ0(s)z(s)dW0(s) minus qp 1113946

t

0σ0(s)dW0(s)1113883

(A9)

Substituting (A8) and (A9) into (A5) leads to


11138681113868111386811138681113868111386811138681113868q

D2(t)1113868111386811138681113868

1113868111386811138681113868qexp1113882qpTr0 + qz(0)I0 + qp 1113946

t

0a(s)(T minus s)ds

+ q 1113946t

0β(s)α(s)z(s)ds1113883

middot exp1113882qp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t

0σ1(s)πlowast1(s)1113858

+ b(T minus s)]dW1(s) + qp 1113946t


+ q 1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859dW0(s)1113883

(A10)


Let

M(t) exp minus12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds minus

12q2p2

1113946t

0σ1(s)πlowast1(s) + b(T minus s)1113858 1113859

2ds1113896

minus12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds minus12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

minus q2pρ1113946

t

0σ1(s)πlowast1(s) + b(T minus s)1113858 1113859 σ0(s)z(s) minus pσ0(s)1113858 1113859ds1113897

middot expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t

0σ1(s)πlowast1(s) + b(T minus s)1113858 1113859dW1(s)1113896

+ qp 1113946t

0σ2(s)πlowast2(s)dW2(s) + q 1113946

t


D3(t) D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qpTr0 + qz(0)I0 + qp 1113946

t

0a(s)(T minus s)ds + q 1113946

t

0β(s)α(s)z(s)ds1113896 1113897

middot exp12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds +

12q2p2

1113946t


2ds1113896

+12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds +12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

+q2pρ1113946

t


(A11)

It is easy to see that M(t) tge0 is a martingale and


11138681113868111386811138681113868111386811138681113868q

D3(t)M(t) (A12)

)us by the optional stopping time theorem and the factthat D3(t) is deterministic and continuous on the interval[0 T] we obtain that for all stopping times τn with 0le τn leT

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875 E D3 τn( 1113857 middot M τn( 11138571113858 1113859

le suptisin[0T]

D3(t) middot E M τn( 11138571113858 1113859le suptisin[0T]

D3(t)ltinfin

(A13)

Note that suptisin[0T]D3(t)ltinfin is independent of n )ustaking supremum over n isin N on both sides of the aboveequation yields

supnisinN

E G τn Xτn r τn( 1113857 I τn( 11138571113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A14)

)erefore uniformly integrable property follows up

A2 Vasicek Model For the Vasicek model we also need toprove that for any fixed qgt 1

supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A15)

Combining equations (43) and (A2) we have for everyfixed qgt 1

G t Xlowastt rt It( 1113857

11138681113868111386811138681113868111386811138681113868q

1


T

t

1113957h(s)ds + q1113957k(t)r(t) + q1113957z(t)I(t)1113896 1113897 Xlowastt( 1113857

pq

1113957D2(t)1113868111386811138681113868

1113868111386811138681113868q exp q1113957k(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897

times exp q1113957z(t)I(t) minus qp 1113946t


t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A16)


where we let

1113957D2(t) 1pexp minus p 1113946

T

t

1113957h(s)ds1113896 1113897Dp1(t) (A17)

Note that 1113957z(t) in the Vasicekmodel is equal to z(t) in theHondashLee model and so the difference between the expressionof |G(t Xlowastt rt It)|

q in the Vasicek model and HondashLee modelis the first part of the expression of |G(t Xlowastt rt It)|

qSince in the Vasicek model r(t) is described by

dr(t) [θ(t) minus 1113954br(t) + bξ(t)]dt + bdW1(t) (A18)

and 1113957k(t) (p1113954b)[1 minus eb1113954(tminus T)] after some calculations wecan easily get

exp q1113957k(t)r(t) + qp 1113946t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897

exp1113882 minusqp

1113954be

minus 1113954bTr0 minus

qp

1113954b1113946

t

0e

b1113954(sminus T)[θ(s) + bξ(s)]ds +

qp

1113954br0

+qp

1113954b1113946

t

0[θ(s) + bξ(s)]ds1113883

times exp 1113946t

0

b

1113954b+ qpσ1(s)πlowast1(s) minus

qp

1113954be

b1113954(sminus T)b1113890 1113891dW1(s)1113896 1113897

(A19)

Now following the same method used in the HondashLeemodel we can finally find a deterministic and continuousfunction 1113957D3(t) and a martingale 1113957M(t) such that


11138681113868111386811138681113868111386811138681113868q

1113957D3(t) 1113957M(t) (A20)

Similar to the method used in the HondashLee model we canobtain the uniformly integrable property ofG(τn Xlowastτn

r(τn) I(τn))1113966 1113967nisinN

)us we complete our proof

Data Availability

)e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to the writing of this paperAll authors read and approved the final version

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (grant nos 11771079 and 11371020)

References

[1] H Markowitz ldquoPortfolio selectionrdquo Be Journal of Financevol 7 no 1 pp 77ndash91 1952

[2] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time caserdquo Be Review of Economics andStatistics vol 51 no 3 pp 247ndash257 1969

[3] R C Merton ldquoOptimum consumption and portfolio rules ina continuous-time modelrdquo Journal of EconomicBeory vol 3no 4 pp 373ndash413 1971

[4] T K Siu ldquoLong-term strategic asset allocation with inflationrisk and regime switchingrdquo Quantitative Finance vol 11no 10 pp 1565ndash1580 2011

[5] C RobertContinuous-Time Finance Blackwell Oxford UK 1990[6] K Ralf Optimal Portfolios Stochastic Models for Optimal

Investment and Risk Management in Continuous Time WorldScientific Singapore 1997

[7] I Karatzas and E Steven Shreve Methods of MathematicalFinance Springer New York NY USA 1998

[8] R J Elliott and P E KoppMathematics of Financial MarketsSpringer Finance New York NY USA 2005

[9] M H A Davis and A R Norman ldquoPortfolio selection withtransaction costsrdquo Mathematics of Operations Researchvol 15 no 4 pp 676ndash713 1990

[10] J F Cocco F J Gomes and P J Maenhout ldquoConsumptionand portfolio choice over the life cyclerdquo Review of FinancialStudies vol 18 no 2 pp 491ndash533 2005

[11] H Kraft ldquoOptimal portfolios and Hestonrsquos stochastic vola-tility model an explicit solution for power utilityrdquo Quanti-tative Finance vol 5 no 3 pp 303ndash313 2005

[12] J Y Campbell and L M Viceira Strategic Asset AllocationPortfolio Choice for Long-Term Investors Clarendon Lecturesin Economics Oxford University Press Oxford UK 2002

[13] A D Wilkie ldquoA stochastic investment model for actuarialuserdquo Transactions of the Faculty of Actuaries vol 39pp 341ndash403 1984

[14] N D Pearson and T-S Sun ldquoExploiting the conditionaldensity in estimating the term structure an application to theCox Ingersoll and Ross modelrdquo Be Journal of Financevol 49 no 4 pp 1279ndash1304 1994

[15] C Munk C Soslashrensen and T Nygaard Vinther ldquoDynamicasset allocation under mean-reverting returns stochasticinterest rates and inflation uncertaintyrdquo International Reviewof Economics amp Finance vol 13 no 2 pp 141ndash166 2004

[16] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquoBe Journal of Finance vol 57 no 3 pp 1201ndash12382002

[17] R Korn T K Siu and A Zhang ldquoAsset allocation for a DCpension fund under regime switching environmentrdquo Euro-pean Actuarial Journal vol 1 no S2 pp 361ndash377 2011

[18] T K Siu ldquoA BSDE approach to risk-based asset allocation ofpension funds with regime switchingrdquo Annals of OperationsResearch vol 2012 no 1 pp 449ndash473 2012

[19] R Korn and H Kraft ldquoA stochastic control approach toportfolio problems with stochastic interest ratesrdquo SIAM Journalon Control and Optimization vol 40 no 4 pp 1250ndash12692002

[20] J Li and R Wu ldquoOptimal investment problem with stochasticinterest rate and stochastic volatility maximizing a powerutilityrdquo Applied Stochastic Models in Business and Industryvol 25 no 3 pp 407ndash420 2009

[21] H Kraft ldquoOptimal portfolios with stochastic short ratepitfalls when the short rate is non-gaussian or themarket priceof risk is unboundedrdquo International Journal ofBeoretical andApplied Finance vol 12 no 6 pp 767ndash796 2009

[22] Y Shen and T K Siu ldquoAsset allocation under stochasticinterest rate with regime switchingrdquo Economic Modellingvol 29 no 4 pp 1126ndash1136 2012


[23] Z Wang J Xia and L Zhang ldquoOptimal investment for aninsurer the martingale approachrdquo Insurance Mathematicsand Economics vol 40 no 2 pp 322ndash334 2007

[24] X Zeng ldquoA stochastic differential reinsurance gamerdquo Journalof Applied Probability vol 47 no 2 pp 335ndash349 2010

[25] S Luo M Taksar and A Tsoi ldquoOn reinsurance and in-vestment for large insurance portfoliosrdquo Insurance Mathe-matics and Economics vol 42 no 1 pp 434ndash444 2008

[26] R J Elliott and T K Siu ldquoA BSDE approach to a risk-basedoptimal investment of an insurerrdquo Automatica vol 47 no 2pp 253ndash261 2011

[27] Z Liang K C Yuen and K C Cheung ldquoOptimal re-insurance-investment problem in a constant elasticity ofvariance stock market for jump-diffusion risk modelrdquoAppliedStochastic Models in Business and Industry vol 28 no 6pp 585ndash597 2012

[28] R J Elliott and T K Siu ldquoAn HMM approach for optimalinvestment of an insurerrdquo International Journal of Robust andNonlinear Control vol 22 no 7 pp 778ndash807 2011

[29] T K Siu ldquoA BSDE approach to optimal investment of aninsurer with hidden regime switchingrdquo Stochastic Analysisand Applications vol 31 no 1 pp 1ndash18 2013

[30] L Bai and H Zhang ldquoDynamic mean-variance problem withconstrained risk control for the insurersrdquo MathematicalMethods of Operations Research vol 68 no 1 pp 181ndash2052008

[31] X Zhang and T K Siu ldquoOptimal investment and reinsuranceof an insurer with model uncertaintyrdquo Insurance Mathe-matics and Economics vol 45 no 1 pp 81ndash88 2009

[32] X Zhang and T K Siu ldquoOn optimal proportional reinsuranceand investment in a Markovian regime-switching economyrdquoActa Mathematica Sinica English Series vol 28 no 1pp 67ndash82 2012

[33] X Zhang R J Elliott and T K Siu ldquoA Bayesian approach foroptimal reinsurance and investment in a diffusion modelrdquoJournal of Engineering Mathematics vol 76 no 1 pp 195ndash206 2012

[34] J Liu K-F Cedric Yiu T K Siu and W-K Ching ldquoOptimalinvestment-reinsurance with dynamic risk constraint andregime switchingrdquo Scandinavian Actuarial Journal vol 2013no 4 pp 263ndash285 2013

[35] T B Douglas ldquoConsumption production inflation and in-terest ratesrdquo Journal of Financial Economics vol 16 no 1pp 3ndash39 1986

[36] G George ldquoIdentifying the dynamics of real interest rates andinflation Evidence using survey datardquo Review of FinancialStudies vol 4 no 1 pp 53ndash86 1991

[37] E F Fama and M R Gibbons ldquoInflation real returns andcapital investmentrdquo Journal of Monetary Economics vol 9no 3 pp 297ndash323 1982

[38] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under in-flation a continuous-time modelrdquo Insurance Mathematicsand Economics vol 53 no 3 pp 851ndash863 2013

[39] G Jan Aspects of Risk Beory Springer New York NY USA1991

[40] Y Zeng and Z Li ldquoOptimal time-consistent investment andreinsurance policies for mean-variance insurersrdquo InsuranceMathematics and Economics vol 49 no 1 pp 145ndash154 2011

[41] B Hoslashjgaard and M Taksar ldquoOptimal proportional re-insurance policies for diffusion modelsrdquo Scandinavian Ac-tuarial Journal vol 1998 no 2 pp 166ndash180 1998

[42] I T Michael and C Markussen ldquoOptimal dynamic re-insurance policies for large insurance portfoliosrdquo Finance andStochastics vol 7 no 1 pp 97ndash121 2003

[43] I Karatzas and S E Shreve Brownian Motion and StochasticCalculus Springer New York NY USA 1991

[44] F Wendell Deterministic and Stochastic Optimal ControlR W Rishel and F Wendell Eds Springer New York NYUSA 1975

[45] H F Wendell ldquoControlled markov processes and viscositysolutionsrdquo in Stochastic Modelling and Applied ProbabilityF Wendell and H M Soner Eds Springer New York NYUSA 2nd edition 2006

[46] T Bui X Cheng Z Jin and G Yin ldquoApproximation of a classof non-zero-sum investment and reinsurance games for re-gime-switching jump-diffusion modelsrdquo Nonlinear AnalysisHybrid Systems vol 32 pp 276ndash293 2019

[47] N Wang N Zhang Z Jin and L Qian ldquoRobust non-zero-sum investment and reinsurance game with default riskrdquoInsurance Mathematics and Economics vol 84 pp 115ndash1322019


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Submit your manuscripts atwwwhindawicom

problem of the insurance company is obtained in each of thetwo stochastic interest rate models namely the HondashLeemodel and the Vasicek model )e verification theorems forthe HJB solutions to the optimal investment and reinsuranceproblem for the two stochastic interest rate models areprovided Numerical results based on hypothetical parametervalues are presented to illustrate the impacts of the twodifferent stochastic interest rate modeling assumptions onoptimal decision making of the insurer In what follows abrief review on some relevant literature and the motivation ofthis study are provided

)e optimal investment-reinsurance problem is one ofthe classic research topics in actuarial and insurancemathematics It aims to provide a theoretical basis for ra-tional decision making of insurers on investments in a se-curity market and transferring insurance risks toreinsurance companies )e rational decisions of the in-surers are made by optimizing certain objective criteria suchas utility function )is is related to the utility maximizationproblem in neoclassical economics as well as financialeconomics )e solution of the optimization problem is byits nature normative though results of positive nature mayoccasionally obtain In financial economics a ldquoformalrdquo studyof an optimal investment problem may be tracked back tothe pioneering work by Markowitz [1] where the scientificfoundation of the problem was enacted by making use ofmathematics Specifically the problem was formulatedmathematically as an optimization problem in a static one-period model where only the expected return of a portfolioand the risk of the portfolio measured by the variance of itsreturn were relevant More realistic situations such ascontinuous-time situations were considered in the secondstage of developments pioneered by the seminal articles byMerton [2 3] In particular the continuous-time optimalasset allocation model by Merton [2 3] brought the his-torical developments of the field in a new era and stimulatedthe development of continuous-time finance (see for ex-ample Siu [4] for related discussion) Since the seminalworks by Merton [2 3] there have been many significantdevelopments in continuous-time optimal asset allocationmodels and a large volume of literature on the field It doesnot seem to be easy to list all of them here but some of themare mentioned without slighting the other literature Onemay refer to the monographs by Merton [5] Ralf [6]Karatzas and Shreve [7] Elliott and Kopp [8] and therelevant literature therein for more detailed exposition onthe developments of the literature in the field Davis andNorman [9] investigated an optimal consumption and in-vestment decision for an investor who invests in a simplifiedsecurity market with a bank account and a stock Fur-thermore the authors took fixed percentage transactioncosts into consideration A life-cycle model of consumptionand portfolio was considered by Cocco et al [10] Kraft [11]studied an optimal portfolio selection problem with Hes-tonrsquos stochastic volatility

Long-term strategic asset allocation is one of the im-portant areas in optimal asset allocation It sheds light onoptimal investment decision making in financial planninginsurance and pension funds Intuitively factors such as

inflation interest rate investment cycle and economic cyclewhich seem to be relevant to long-term investment decisionmaking may be taken into account when discussing long-term strategic asset allocation )e monograph by Campbelland Viceira [12] provides an excellent account of long-termstrategic asset allocation where the problem was rigorouslytreated in discrete time modeling frameworks)ere is also aconsiderable amount of finance and insurance literature onlong-term strategic asset allocation Again some literaturestudies particularly those relating to incorporation of in-flation and interest rate are mentioned here Inflation has asignificant impact on long-term investment decision mak-ing Indeed inflation was used as the main driver of in-vestment return series in long run in one of the majormodels in actuarial science namely the Wilkie stochasticinvestment model developed by Wilkie [13] A key featureof inflation namely mean reversion was described by anautoregressive time series model (in discrete time) in theWilkie stochastic investmentmodel (see for example Siu [4]for related discussion) Pearson and Sun [14] described theinflation risk by a mean-reverting square-root process whilein Munk et al [15] and Brennan and Xia [16] the inflationprocess is modeled by an OrnsteinndashUhlenbeck process Siu[4] discussed a long-term strategic asset allocation problemin continuous time by incorporating inflation and regimeswitching Korn et al [17] and Siu [18] considered an op-timal asset allocation of pension funds in the presence ofboth the inflation and regime switching risks using amartingale approach and a backward stochastic differentialequation approach respectively See also the relevant lit-erature in the papers by Siu [4] Korn et al [17] and Siu [18]for some works on optimal asset allocation in the presence ofinflation Apart from inflation risk the interest risk is an-other important factor that we have to face Korn and Kraft[19] studied the portfolio problems with stochastic interestrates In Li and Wu [20] stochastic interest rate is given bythe CoxndashIngersollndashRoss (CIR) model and the volatility of thestock is also a CIR process Kraft [21] considered all commonshort rate models and stochastic discount Shen and Siu [22]investigated an optimal asset allocation problem in thepresence of both stochastic interest rate and regimeswitching effect where a regime-switching version of theVasicek model was used to describe stochastic interest rate

)ere has been a large amount of work on optimalinvestment of an insurance company Some of the literaturestudies are for example Wang et al [23] Zeng [24] Luoet al [25] Elliott and Siu [26] Liang et al [27] Elliott and Siu[28] and Siu [29] For an insurance company it is natural toadopt a prudent investment strategy that is they should notonly diversify risk by investing their money into differentassets but also should avoid the risk brought from thecompanyrsquos relatively long investment cycle Over a longperiod of time accumulated inflation can lead to hugeshrinkage of the wealth What is more relevant here is acombined optimal investment and reinsurance problem ofan insurer Some of the literature is also mentioned here Baiand Zhang [30] considered an optimal investment-re-insurance problem for an insurer under the mean-variancecriterion Zhang and Siu [31] incorporated model









dΠ(t) Π(t) I(t)dt + σ0(t)dW0(t)1113858 1113859








dB(t) r(t)B(t)dt B(0) 1 (3)







P 0 T1( 1113857 P0 gt 0

(5)







RP

(7)














d 1113957X(t) π1(t) 1113957X(t)dP(t)

P(t)+ π2(t) 1113957X(t)

dS(t)

S(t)





+ π1(t)σ1(t)dW1(t) + π2(t)σ2(t)dW2(t)1113883


(11)








(12)




+ λ(t)σ2(t)π2(t)1113877 + u(t)c(t)1113883dt



(13)







Vπ1113957(t x r I) E1p

Xπ1113957T1113874 1113875

p1113868111386811138681113868111386811138681113868Xt x rt r It I1113896 1113897 (14)








⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(15)



Vπ1113957(t x r I) (16)






20(t)x

2+ a(t)Vr




111388212Vxxσ

21(t)x

2π21 + 1113876Vxσ1(t)η(t)x


+ Vxrbσ1(t)x

+ VxIσ0(t)σ1(t)ρx1113877π11113883

+ supπ2isinR

12Vxxσ

22(t)x

2π22 + Vxλ(t)σ2(t)xπ21113882 1113883

+ supuisin[0infin)

12Vxxσ

23(t)u

2+ c(t)Vxu1113882 1113883

(18)

V(T x I r) 1p

xp (19)



p (20)




σ1(t)

Gx

xGxx

minusσ0(t)ρσ1(t)

GIx

xGxx

minusb

σ1(t)

Gxr

xGxx

+σ0(t)ρσ1(t)

(21)


xGxx

λ(t)

σ2(t) (22)

ulowast(t) minus

Gx

Gxx

c(t)

σ23(t) (23)


Gt gt

xp

p

Gx gxpminus 1


Gr gr

xp

p

Grr grr

xp

p

GI gI

xp

p

GII gII

xp

p

Gxr grxpminus 1

GxI gIxpminus 1

GrI grI

xp

p

(24)


0 gt

p+ g r + σ20 minus I1113872 1113873 +


1p

+12b2grr

p

+gI

pβ(α minus I) +

12σ20

gII


gIr

pbρσ0

minusg

p minus 1η2

2minus12

g(p minus 1)σ20ρ2

minusg2

I

g(p minus 1)

σ20ρ2

2

minusg2

r

g(p minus 1)

b2

2+ gησ0ρ minus

gI


gr

p minus 1bη


minusgrgI


g

p minus 1λ2

2

minusg

p minus 1c2

2σ23

(25)



(26)



gt fprimeeΔ + feΔ


gr fkeΔ

gI fzeΔ

gIr fkzeΔ

grr fk2eΔ

gII fz2eΔ

(27)


0 kprimep

+ 11113890 1113891fr +zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ 1113882σ20 +12


2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz minus

λ2

2(p minus 1)

minusc2

2σ23(p minus 1)1113883f +

ak

pf

(28)



h(t) ≔ σ20 +12

(p minus 1)σ20 +(1113957a + bξ)k

p+

b2k2

2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz

minusλ2

2(p minus 1)minus

c2

2σ23(p minus 1)

(29)


kprime(t) minus p

k(T) 0

⎧⎨

⎩ (30)


z(T) 01113896 (31)


f(T) 11113896 (32)


0 kprimep

+ 11113890 1113891fr +zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ h(t)f (33)



k(t) p(T minus t)


t

0β(s)ds

1113946

T

t

eminus 1113938

s

0β(v)dvds


T

th(s)ds

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)



T

th(s)ds + k(t)r + z(t)I1113896 1113897x

p

(35)



σ1(t)

1p minus 1

minusb

σ1(t)

p


ρσ0(t)

σ1(t)

minusρσ0(t)

σ1(t)

p

p minus 1e1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds


σ2(t)

1p minus 1

ulowast(t) minus

c(t)

σ23(t)

1p minus 1

x

(36)



1113957h(t) ≔ σ20 +12


p+

b2k2

2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz minus

λ2

2(p minus 1)

minusc2

2σ23(p minus 1)

(37)



1113957k(T) 0

⎧⎨

⎩ (38)


1113957z(T) 01113896 (39)

1113957fprime(t) + p1113957h(t)1113957f(t) 0

1113957f(T) 11113896 (40)


0 kprimep

minus1113954b

pk + 11113890 1113891fr +

zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ 1113957hf (41)



1113957k(t) p


1113957z(t) minus pe1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds

1113957f(t) eminus p 1113938

T

t

~h(s)ds

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)



T

t

1113957h(s)ds + 1113957k(t)r + 1113957z(t)I1113896 1113897xp

(43)



σ1(t)

1p minus 1

minus1

σ1(t)

p

p minus 11 minus e

b(tminus T)1113960 1113961

minusρσ0(t)

σ1(t)

p

p minus 1e1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds +

ρσ0(t)

σ1(t)


σ2(t)

1p minus 1


σ23(t)

1p minus 1

x

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)











I(τn) r(τn))1113966 1113967nisinN is






EtxrI

1p

Xπ1113957T1113874 1113875

p

1113890 1113891leG(t x r I) (45)


V(t x r I) EtxrI

1p

Xπ1113957lowastT1113874 1113875

p

1113890 1113891 G(t x r I) (46)


EtxrI

1p

Xπ1113957T1113874 1113875

p

1113890 1113891leG(t x r I) (47)


V(t x r I) EtxrI

1p

Xπ1113957lowastT1113874 1113875

p

1113890 1113891 G(t x r I) (48)







p = 04p = 05p = 06

0 10 20 30 40 50 60 70 80minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0


0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

p = 04p = 05p = 06




6 Conclusion


Appendix



dXlowastt X



σ23(t)

11 minus p

1113877dt +c(t)

σ3(t)

11 minus p

dW3(t)



(A1)



t


t

0I(s)ds

+1

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + 1113946

t


+ 1113946t


t

0σ0(s)dW0(s)1113883

(A2)

where

D1(t) X0 exp1113946t


+ σ20(s) +c2(s)


c2(s)



2


2

2minusσ20(s)

2


(A3)




any fixed qgt 1

0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0




supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q


fixed qgt 1


11138681113868111386811138681113868111386811138681113868q

1


T


lowastt( 1113857

pq

D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qk(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A5)

where we set


T

th(s)ds1113896 1113897D

p1(t) (A6)


0 sdr(s) + 1113938t



0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



+ qp 1113946t


(A7)


z(t)I(t) z(0)I0 + 1113946t


t

0z(s)dI(s)

z(0)I0 + 1113946t


t

0z(s) β(s)[α(s)1113864


z(0)I0 + 1113946t

0pI(s)ds + 1113946

t

0z(s)β(s)α(s)ds

+ 1113946t

0σ0(s)z(s)dW0(s)

(A8)




t

0σ0(s)dW0(s)1113896 1113897

exp1113882qz(0)I0 + q 1113946t

0β(s)α(s)z(s)ds

+ q 1113946t


t

0σ0(s)dW0(s)1113883

(A9)



11138681113868111386811138681113868111386811138681113868q

D2(t)1113868111386811138681113868

1113868111386811138681113868qexp1113882qpTr0 + qz(0)I0 + qp 1113946

t

0a(s)(T minus s)ds

+ q 1113946t

0β(s)α(s)z(s)ds1113883

middot exp1113882qp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t




+ q 1113946t


(A10)


Let

M(t) exp minus12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds minus

12q2p2

1113946t


2ds1113896

minus12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds minus12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

minus q2pρ1113946

t


middot expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t


+ qp 1113946t


t


D3(t) D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qpTr0 + qz(0)I0 + qp 1113946

t


t

0β(s)α(s)z(s)ds1113896 1113897

middot exp12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds +

12q2p2

1113946t


2ds1113896

+12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds +12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

+q2pρ1113946

t


(A11)



11138681113868111386811138681113868111386811138681113868q

D3(t)M(t) (A12)


E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875 E D3 τn( 1113857 middot M τn( 11138571113858 1113859

le suptisin[0T]


D3(t)ltinfin

(A13)


supnisinN

E G τn Xτn r τn( 1113857 I τn( 11138571113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A14)



supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A15)



11138681113868111386811138681113868111386811138681113868q

1


T

t


pq

1113957D2(t)1113868111386811138681113868

1113868111386811138681113868q exp q1113957k(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A16)


where we let

1113957D2(t) 1pexp minus p 1113946

T

t

1113957h(s)ds1113896 1113897Dp1(t) (A17)






exp q1113957k(t)r(t) + qp 1113946t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897

exp1113882 minusqp

1113954be


qp

1113954b1113946

t

0e


qp

1113954br0

+qp

1113954b1113946

t

0[θ(s) + bξ(s)]ds1113883

times exp 1113946t

0

b


qp

1113954be

b1113954(sminus T)b1113890 1113891dW1(s)1113896 1113897

(A19)



11138681113868111386811138681113868111386811138681113868q

1113957D3(t) 1113957M(t) (A20)


r(τn) I(τn))1113966 1113967nisinN


Data Availability






Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018















dΠ(t) Π(t) I(t)dt + σ0(t)dW0(t)1113858 1113859








dB(t) r(t)B(t)dt B(0) 1 (3)







P 0 T1( 1113857 P0 gt 0

(5)







RP

(7)














d 1113957X(t) π1(t) 1113957X(t)dP(t)

P(t)+ π2(t) 1113957X(t)

dS(t)

S(t)





+ π1(t)σ1(t)dW1(t) + π2(t)σ2(t)dW2(t)1113883


(11)








(12)




+ λ(t)σ2(t)π2(t)1113877 + u(t)c(t)1113883dt



(13)







Vπ1113957(t x r I) E1p

Xπ1113957T1113874 1113875

p1113868111386811138681113868111386811138681113868Xt x rt r It I1113896 1113897 (14)








⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(15)



Vπ1113957(t x r I) (16)






20(t)x

2+ a(t)Vr




111388212Vxxσ

21(t)x

2π21 + 1113876Vxσ1(t)η(t)x


+ Vxrbσ1(t)x

+ VxIσ0(t)σ1(t)ρx1113877π11113883

+ supπ2isinR

12Vxxσ

22(t)x

2π22 + Vxλ(t)σ2(t)xπ21113882 1113883

+ supuisin[0infin)

12Vxxσ

23(t)u

2+ c(t)Vxu1113882 1113883

(18)

V(T x I r) 1p

xp (19)



p (20)




σ1(t)

Gx

xGxx

minusσ0(t)ρσ1(t)

GIx

xGxx

minusb

σ1(t)

Gxr

xGxx

+σ0(t)ρσ1(t)

(21)


xGxx

λ(t)

σ2(t) (22)

ulowast(t) minus

Gx

Gxx

c(t)

σ23(t) (23)


Gt gt

xp

p

Gx gxpminus 1


Gr gr

xp

p

Grr grr

xp

p

GI gI

xp

p

GII gII

xp

p

Gxr grxpminus 1

GxI gIxpminus 1

GrI grI

xp

p

(24)


0 gt

p+ g r + σ20 minus I1113872 1113873 +


1p

+12b2grr

p

+gI

pβ(α minus I) +

12σ20

gII


gIr

pbρσ0

minusg

p minus 1η2

2minus12

g(p minus 1)σ20ρ2

minusg2

I

g(p minus 1)

σ20ρ2

2

minusg2

r

g(p minus 1)

b2

2+ gησ0ρ minus

gI


gr

p minus 1bη


minusgrgI


g

p minus 1λ2

2

minusg

p minus 1c2

2σ23

(25)



(26)



gt fprimeeΔ + feΔ


gr fkeΔ

gI fzeΔ

gIr fkzeΔ

grr fk2eΔ

gII fz2eΔ

(27)


0 kprimep

+ 11113890 1113891fr +zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ 1113882σ20 +12


2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz minus

λ2

2(p minus 1)

minusc2

2σ23(p minus 1)1113883f +

ak

pf

(28)



h(t) ≔ σ20 +12

(p minus 1)σ20 +(1113957a + bξ)k

p+

b2k2

2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz

minusλ2

2(p minus 1)minus

c2

2σ23(p minus 1)

(29)


kprime(t) minus p

k(T) 0

⎧⎨

⎩ (30)


z(T) 01113896 (31)


f(T) 11113896 (32)


0 kprimep

+ 11113890 1113891fr +zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ h(t)f (33)



k(t) p(T minus t)


t

0β(s)ds

1113946

T

t

eminus 1113938

s

0β(v)dvds


T

th(s)ds

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)



T

th(s)ds + k(t)r + z(t)I1113896 1113897x

p

(35)



σ1(t)

1p minus 1

minusb

σ1(t)

p


ρσ0(t)

σ1(t)

minusρσ0(t)

σ1(t)

p

p minus 1e1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds


σ2(t)

1p minus 1

ulowast(t) minus

c(t)

σ23(t)

1p minus 1

x

(36)



1113957h(t) ≔ σ20 +12


p+

b2k2

2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz minus

λ2

2(p minus 1)

minusc2

2σ23(p minus 1)

(37)



1113957k(T) 0

⎧⎨

⎩ (38)


1113957z(T) 01113896 (39)

1113957fprime(t) + p1113957h(t)1113957f(t) 0

1113957f(T) 11113896 (40)


0 kprimep

minus1113954b

pk + 11113890 1113891fr +

zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ 1113957hf (41)



1113957k(t) p


1113957z(t) minus pe1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds

1113957f(t) eminus p 1113938

T

t

~h(s)ds

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)



T

t

1113957h(s)ds + 1113957k(t)r + 1113957z(t)I1113896 1113897xp

(43)



σ1(t)

1p minus 1

minus1

σ1(t)

p

p minus 11 minus e

b(tminus T)1113960 1113961

minusρσ0(t)

σ1(t)

p

p minus 1e1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds +

ρσ0(t)

σ1(t)


σ2(t)

1p minus 1


σ23(t)

1p minus 1

x

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)











I(τn) r(τn))1113966 1113967nisinN is






EtxrI

1p

Xπ1113957T1113874 1113875

p

1113890 1113891leG(t x r I) (45)


V(t x r I) EtxrI

1p

Xπ1113957lowastT1113874 1113875

p

1113890 1113891 G(t x r I) (46)


EtxrI

1p

Xπ1113957T1113874 1113875

p

1113890 1113891leG(t x r I) (47)


V(t x r I) EtxrI

1p

Xπ1113957lowastT1113874 1113875

p

1113890 1113891 G(t x r I) (48)







p = 04p = 05p = 06

0 10 20 30 40 50 60 70 80minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0


0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

p = 04p = 05p = 06




6 Conclusion


Appendix



dXlowastt X



σ23(t)

11 minus p

1113877dt +c(t)

σ3(t)

11 minus p

dW3(t)



(A1)



t


t

0I(s)ds

+1

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + 1113946

t


+ 1113946t


t

0σ0(s)dW0(s)1113883

(A2)

where

D1(t) X0 exp1113946t


+ σ20(s) +c2(s)


c2(s)



2


2

2minusσ20(s)

2


(A3)




any fixed qgt 1

0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0




supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q


fixed qgt 1


11138681113868111386811138681113868111386811138681113868q

1


T


lowastt( 1113857

pq

D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qk(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A5)

where we set


T

th(s)ds1113896 1113897D

p1(t) (A6)


0 sdr(s) + 1113938t



0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



+ qp 1113946t


(A7)


z(t)I(t) z(0)I0 + 1113946t


t

0z(s)dI(s)

z(0)I0 + 1113946t


t

0z(s) β(s)[α(s)1113864


z(0)I0 + 1113946t

0pI(s)ds + 1113946

t

0z(s)β(s)α(s)ds

+ 1113946t

0σ0(s)z(s)dW0(s)

(A8)




t

0σ0(s)dW0(s)1113896 1113897

exp1113882qz(0)I0 + q 1113946t

0β(s)α(s)z(s)ds

+ q 1113946t


t

0σ0(s)dW0(s)1113883

(A9)



11138681113868111386811138681113868111386811138681113868q

D2(t)1113868111386811138681113868

1113868111386811138681113868qexp1113882qpTr0 + qz(0)I0 + qp 1113946

t

0a(s)(T minus s)ds

+ q 1113946t

0β(s)α(s)z(s)ds1113883

middot exp1113882qp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t




+ q 1113946t


(A10)


Let

M(t) exp minus12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds minus

12q2p2

1113946t


2ds1113896

minus12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds minus12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

minus q2pρ1113946

t


middot expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t


+ qp 1113946t


t


D3(t) D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qpTr0 + qz(0)I0 + qp 1113946

t


t

0β(s)α(s)z(s)ds1113896 1113897

middot exp12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds +

12q2p2

1113946t


2ds1113896

+12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds +12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

+q2pρ1113946

t


(A11)



11138681113868111386811138681113868111386811138681113868q

D3(t)M(t) (A12)


E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875 E D3 τn( 1113857 middot M τn( 11138571113858 1113859

le suptisin[0T]


D3(t)ltinfin

(A13)


supnisinN

E G τn Xτn r τn( 1113857 I τn( 11138571113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A14)



supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A15)



11138681113868111386811138681113868111386811138681113868q

1


T

t


pq

1113957D2(t)1113868111386811138681113868

1113868111386811138681113868q exp q1113957k(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A16)


where we let

1113957D2(t) 1pexp minus p 1113946

T

t

1113957h(s)ds1113896 1113897Dp1(t) (A17)






exp q1113957k(t)r(t) + qp 1113946t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897

exp1113882 minusqp

1113954be


qp

1113954b1113946

t

0e


qp

1113954br0

+qp

1113954b1113946

t

0[θ(s) + bξ(s)]ds1113883

times exp 1113946t

0

b


qp

1113954be

b1113954(sminus T)b1113890 1113891dW1(s)1113896 1113897

(A19)



11138681113868111386811138681113868111386811138681113868q

1113957D3(t) 1113957M(t) (A20)


r(τn) I(τn))1113966 1113967nisinN


Data Availability






Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018









P 0 T1( 1113857 P0 gt 0

(5)







RP

(7)














d 1113957X(t) π1(t) 1113957X(t)dP(t)

P(t)+ π2(t) 1113957X(t)

dS(t)

S(t)





+ π1(t)σ1(t)dW1(t) + π2(t)σ2(t)dW2(t)1113883


(11)








(12)




+ λ(t)σ2(t)π2(t)1113877 + u(t)c(t)1113883dt



(13)







Vπ1113957(t x r I) E1p

Xπ1113957T1113874 1113875

p1113868111386811138681113868111386811138681113868Xt x rt r It I1113896 1113897 (14)








⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(15)



Vπ1113957(t x r I) (16)






20(t)x

2+ a(t)Vr




111388212Vxxσ

21(t)x

2π21 + 1113876Vxσ1(t)η(t)x


+ Vxrbσ1(t)x

+ VxIσ0(t)σ1(t)ρx1113877π11113883

+ supπ2isinR

12Vxxσ

22(t)x

2π22 + Vxλ(t)σ2(t)xπ21113882 1113883

+ supuisin[0infin)

12Vxxσ

23(t)u

2+ c(t)Vxu1113882 1113883

(18)

V(T x I r) 1p

xp (19)



p (20)




σ1(t)

Gx

xGxx

minusσ0(t)ρσ1(t)

GIx

xGxx

minusb

σ1(t)

Gxr

xGxx

+σ0(t)ρσ1(t)

(21)


xGxx

λ(t)

σ2(t) (22)

ulowast(t) minus

Gx

Gxx

c(t)

σ23(t) (23)


Gt gt

xp

p

Gx gxpminus 1


Gr gr

xp

p

Grr grr

xp

p

GI gI

xp

p

GII gII

xp

p

Gxr grxpminus 1

GxI gIxpminus 1

GrI grI

xp

p

(24)


0 gt

p+ g r + σ20 minus I1113872 1113873 +


1p

+12b2grr

p

+gI

pβ(α minus I) +

12σ20

gII


gIr

pbρσ0

minusg

p minus 1η2

2minus12

g(p minus 1)σ20ρ2

minusg2

I

g(p minus 1)

σ20ρ2

2

minusg2

r

g(p minus 1)

b2

2+ gησ0ρ minus

gI


gr

p minus 1bη


minusgrgI


g

p minus 1λ2

2

minusg

p minus 1c2

2σ23

(25)



(26)



gt fprimeeΔ + feΔ


gr fkeΔ

gI fzeΔ

gIr fkzeΔ

grr fk2eΔ

gII fz2eΔ

(27)


0 kprimep

+ 11113890 1113891fr +zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ 1113882σ20 +12


2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz minus

λ2

2(p minus 1)

minusc2

2σ23(p minus 1)1113883f +

ak

pf

(28)



h(t) ≔ σ20 +12

(p minus 1)σ20 +(1113957a + bξ)k

p+

b2k2

2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz

minusλ2

2(p minus 1)minus

c2

2σ23(p minus 1)

(29)


kprime(t) minus p

k(T) 0

⎧⎨

⎩ (30)


z(T) 01113896 (31)


f(T) 11113896 (32)


0 kprimep

+ 11113890 1113891fr +zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ h(t)f (33)



k(t) p(T minus t)


t

0β(s)ds

1113946

T

t

eminus 1113938

s

0β(v)dvds


T

th(s)ds

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)



T

th(s)ds + k(t)r + z(t)I1113896 1113897x

p

(35)



σ1(t)

1p minus 1

minusb

σ1(t)

p


ρσ0(t)

σ1(t)

minusρσ0(t)

σ1(t)

p

p minus 1e1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds


σ2(t)

1p minus 1

ulowast(t) minus

c(t)

σ23(t)

1p minus 1

x

(36)



1113957h(t) ≔ σ20 +12


p+

b2k2

2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz minus

λ2

2(p minus 1)

minusc2

2σ23(p minus 1)

(37)



1113957k(T) 0

⎧⎨

⎩ (38)


1113957z(T) 01113896 (39)

1113957fprime(t) + p1113957h(t)1113957f(t) 0

1113957f(T) 11113896 (40)


0 kprimep

minus1113954b

pk + 11113890 1113891fr +

zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ 1113957hf (41)



1113957k(t) p


1113957z(t) minus pe1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds

1113957f(t) eminus p 1113938

T

t

~h(s)ds

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)



T

t

1113957h(s)ds + 1113957k(t)r + 1113957z(t)I1113896 1113897xp

(43)



σ1(t)

1p minus 1

minus1

σ1(t)

p

p minus 11 minus e

b(tminus T)1113960 1113961

minusρσ0(t)

σ1(t)

p

p minus 1e1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds +

ρσ0(t)

σ1(t)


σ2(t)

1p minus 1


σ23(t)

1p minus 1

x

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)











I(τn) r(τn))1113966 1113967nisinN is






EtxrI

1p

Xπ1113957T1113874 1113875

p

1113890 1113891leG(t x r I) (45)


V(t x r I) EtxrI

1p

Xπ1113957lowastT1113874 1113875

p

1113890 1113891 G(t x r I) (46)


EtxrI

1p

Xπ1113957T1113874 1113875

p

1113890 1113891leG(t x r I) (47)


V(t x r I) EtxrI

1p

Xπ1113957lowastT1113874 1113875

p

1113890 1113891 G(t x r I) (48)







p = 04p = 05p = 06

0 10 20 30 40 50 60 70 80minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0


0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

p = 04p = 05p = 06




6 Conclusion


Appendix



dXlowastt X



σ23(t)

11 minus p

1113877dt +c(t)

σ3(t)

11 minus p

dW3(t)



(A1)



t


t

0I(s)ds

+1

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + 1113946

t


+ 1113946t


t

0σ0(s)dW0(s)1113883

(A2)

where

D1(t) X0 exp1113946t


+ σ20(s) +c2(s)


c2(s)



2


2

2minusσ20(s)

2


(A3)




any fixed qgt 1

0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0




supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q


fixed qgt 1


11138681113868111386811138681113868111386811138681113868q

1


T


lowastt( 1113857

pq

D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qk(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A5)

where we set


T

th(s)ds1113896 1113897D

p1(t) (A6)


0 sdr(s) + 1113938t



0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



+ qp 1113946t


(A7)


z(t)I(t) z(0)I0 + 1113946t


t

0z(s)dI(s)

z(0)I0 + 1113946t


t

0z(s) β(s)[α(s)1113864


z(0)I0 + 1113946t

0pI(s)ds + 1113946

t

0z(s)β(s)α(s)ds

+ 1113946t

0σ0(s)z(s)dW0(s)

(A8)




t

0σ0(s)dW0(s)1113896 1113897

exp1113882qz(0)I0 + q 1113946t

0β(s)α(s)z(s)ds

+ q 1113946t


t

0σ0(s)dW0(s)1113883

(A9)



11138681113868111386811138681113868111386811138681113868q

D2(t)1113868111386811138681113868

1113868111386811138681113868qexp1113882qpTr0 + qz(0)I0 + qp 1113946

t

0a(s)(T minus s)ds

+ q 1113946t

0β(s)α(s)z(s)ds1113883

middot exp1113882qp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t




+ q 1113946t


(A10)


Let

M(t) exp minus12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds minus

12q2p2

1113946t


2ds1113896

minus12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds minus12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

minus q2pρ1113946

t


middot expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t


+ qp 1113946t


t


D3(t) D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qpTr0 + qz(0)I0 + qp 1113946

t


t

0β(s)α(s)z(s)ds1113896 1113897

middot exp12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds +

12q2p2

1113946t


2ds1113896

+12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds +12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

+q2pρ1113946

t


(A11)



11138681113868111386811138681113868111386811138681113868q

D3(t)M(t) (A12)


E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875 E D3 τn( 1113857 middot M τn( 11138571113858 1113859

le suptisin[0T]


D3(t)ltinfin

(A13)


supnisinN

E G τn Xτn r τn( 1113857 I τn( 11138571113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A14)



supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A15)



11138681113868111386811138681113868111386811138681113868q

1


T

t


pq

1113957D2(t)1113868111386811138681113868

1113868111386811138681113868q exp q1113957k(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A16)


where we let

1113957D2(t) 1pexp minus p 1113946

T

t

1113957h(s)ds1113896 1113897Dp1(t) (A17)






exp q1113957k(t)r(t) + qp 1113946t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897

exp1113882 minusqp

1113954be


qp

1113954b1113946

t

0e


qp

1113954br0

+qp

1113954b1113946

t

0[θ(s) + bξ(s)]ds1113883

times exp 1113946t

0

b


qp

1113954be

b1113954(sminus T)b1113890 1113891dW1(s)1113896 1113897

(A19)



11138681113868111386811138681113868111386811138681113868q

1113957D3(t) 1113957M(t) (A20)


r(τn) I(τn))1113966 1113967nisinN


Data Availability






Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018













(12)




+ λ(t)σ2(t)π2(t)1113877 + u(t)c(t)1113883dt



(13)







Vπ1113957(t x r I) E1p

Xπ1113957T1113874 1113875

p1113868111386811138681113868111386811138681113868Xt x rt r It I1113896 1113897 (14)








⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(15)



Vπ1113957(t x r I) (16)






20(t)x

2+ a(t)Vr




111388212Vxxσ

21(t)x

2π21 + 1113876Vxσ1(t)η(t)x


+ Vxrbσ1(t)x

+ VxIσ0(t)σ1(t)ρx1113877π11113883

+ supπ2isinR

12Vxxσ

22(t)x

2π22 + Vxλ(t)σ2(t)xπ21113882 1113883

+ supuisin[0infin)

12Vxxσ

23(t)u

2+ c(t)Vxu1113882 1113883

(18)

V(T x I r) 1p

xp (19)



p (20)




σ1(t)

Gx

xGxx

minusσ0(t)ρσ1(t)

GIx

xGxx

minusb

σ1(t)

Gxr

xGxx

+σ0(t)ρσ1(t)

(21)


xGxx

λ(t)

σ2(t) (22)

ulowast(t) minus

Gx

Gxx

c(t)

σ23(t) (23)


Gt gt

xp

p

Gx gxpminus 1


Gr gr

xp

p

Grr grr

xp

p

GI gI

xp

p

GII gII

xp

p

Gxr grxpminus 1

GxI gIxpminus 1

GrI grI

xp

p

(24)


0 gt

p+ g r + σ20 minus I1113872 1113873 +


1p

+12b2grr

p

+gI

pβ(α minus I) +

12σ20

gII


gIr

pbρσ0

minusg

p minus 1η2

2minus12

g(p minus 1)σ20ρ2

minusg2

I

g(p minus 1)

σ20ρ2

2

minusg2

r

g(p minus 1)

b2

2+ gησ0ρ minus

gI


gr

p minus 1bη


minusgrgI


g

p minus 1λ2

2

minusg

p minus 1c2

2σ23

(25)



(26)



gt fprimeeΔ + feΔ


gr fkeΔ

gI fzeΔ

gIr fkzeΔ

grr fk2eΔ

gII fz2eΔ

(27)


0 kprimep

+ 11113890 1113891fr +zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ 1113882σ20 +12


2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz minus

λ2

2(p minus 1)

minusc2

2σ23(p minus 1)1113883f +

ak

pf

(28)



h(t) ≔ σ20 +12

(p minus 1)σ20 +(1113957a + bξ)k

p+

b2k2

2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz

minusλ2

2(p minus 1)minus

c2

2σ23(p minus 1)

(29)


kprime(t) minus p

k(T) 0

⎧⎨

⎩ (30)


z(T) 01113896 (31)


f(T) 11113896 (32)


0 kprimep

+ 11113890 1113891fr +zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ h(t)f (33)



k(t) p(T minus t)


t

0β(s)ds

1113946

T

t

eminus 1113938

s

0β(v)dvds


T

th(s)ds

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)



T

th(s)ds + k(t)r + z(t)I1113896 1113897x

p

(35)



σ1(t)

1p minus 1

minusb

σ1(t)

p


ρσ0(t)

σ1(t)

minusρσ0(t)

σ1(t)

p

p minus 1e1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds


σ2(t)

1p minus 1

ulowast(t) minus

c(t)

σ23(t)

1p minus 1

x

(36)



1113957h(t) ≔ σ20 +12


p+

b2k2

2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz minus

λ2

2(p minus 1)

minusc2

2σ23(p minus 1)

(37)



1113957k(T) 0

⎧⎨

⎩ (38)


1113957z(T) 01113896 (39)

1113957fprime(t) + p1113957h(t)1113957f(t) 0

1113957f(T) 11113896 (40)


0 kprimep

minus1113954b

pk + 11113890 1113891fr +

zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ 1113957hf (41)



1113957k(t) p


1113957z(t) minus pe1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds

1113957f(t) eminus p 1113938

T

t

~h(s)ds

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)



T

t

1113957h(s)ds + 1113957k(t)r + 1113957z(t)I1113896 1113897xp

(43)



σ1(t)

1p minus 1

minus1

σ1(t)

p

p minus 11 minus e

b(tminus T)1113960 1113961

minusρσ0(t)

σ1(t)

p

p minus 1e1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds +

ρσ0(t)

σ1(t)


σ2(t)

1p minus 1


σ23(t)

1p minus 1

x

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)











I(τn) r(τn))1113966 1113967nisinN is






EtxrI

1p

Xπ1113957T1113874 1113875

p

1113890 1113891leG(t x r I) (45)


V(t x r I) EtxrI

1p

Xπ1113957lowastT1113874 1113875

p

1113890 1113891 G(t x r I) (46)


EtxrI

1p

Xπ1113957T1113874 1113875

p

1113890 1113891leG(t x r I) (47)


V(t x r I) EtxrI

1p

Xπ1113957lowastT1113874 1113875

p

1113890 1113891 G(t x r I) (48)







p = 04p = 05p = 06

0 10 20 30 40 50 60 70 80minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0


0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

p = 04p = 05p = 06




6 Conclusion


Appendix



dXlowastt X



σ23(t)

11 minus p

1113877dt +c(t)

σ3(t)

11 minus p

dW3(t)



(A1)



t


t

0I(s)ds

+1

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + 1113946

t


+ 1113946t


t

0σ0(s)dW0(s)1113883

(A2)

where

D1(t) X0 exp1113946t


+ σ20(s) +c2(s)


c2(s)



2


2

2minusσ20(s)

2


(A3)




any fixed qgt 1

0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0




supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q


fixed qgt 1


11138681113868111386811138681113868111386811138681113868q

1


T


lowastt( 1113857

pq

D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qk(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A5)

where we set


T

th(s)ds1113896 1113897D

p1(t) (A6)


0 sdr(s) + 1113938t



0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



+ qp 1113946t


(A7)


z(t)I(t) z(0)I0 + 1113946t


t

0z(s)dI(s)

z(0)I0 + 1113946t


t

0z(s) β(s)[α(s)1113864


z(0)I0 + 1113946t

0pI(s)ds + 1113946

t

0z(s)β(s)α(s)ds

+ 1113946t

0σ0(s)z(s)dW0(s)

(A8)




t

0σ0(s)dW0(s)1113896 1113897

exp1113882qz(0)I0 + q 1113946t

0β(s)α(s)z(s)ds

+ q 1113946t


t

0σ0(s)dW0(s)1113883

(A9)



11138681113868111386811138681113868111386811138681113868q

D2(t)1113868111386811138681113868

1113868111386811138681113868qexp1113882qpTr0 + qz(0)I0 + qp 1113946

t

0a(s)(T minus s)ds

+ q 1113946t

0β(s)α(s)z(s)ds1113883

middot exp1113882qp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t




+ q 1113946t


(A10)


Let

M(t) exp minus12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds minus

12q2p2

1113946t


2ds1113896

minus12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds minus12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

minus q2pρ1113946

t


middot expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t


+ qp 1113946t


t


D3(t) D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qpTr0 + qz(0)I0 + qp 1113946

t


t

0β(s)α(s)z(s)ds1113896 1113897

middot exp12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds +

12q2p2

1113946t


2ds1113896

+12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds +12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

+q2pρ1113946

t


(A11)



11138681113868111386811138681113868111386811138681113868q

D3(t)M(t) (A12)


E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875 E D3 τn( 1113857 middot M τn( 11138571113858 1113859

le suptisin[0T]


D3(t)ltinfin

(A13)


supnisinN

E G τn Xτn r τn( 1113857 I τn( 11138571113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A14)



supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A15)



11138681113868111386811138681113868111386811138681113868q

1


T

t


pq

1113957D2(t)1113868111386811138681113868

1113868111386811138681113868q exp q1113957k(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A16)


where we let

1113957D2(t) 1pexp minus p 1113946

T

t

1113957h(s)ds1113896 1113897Dp1(t) (A17)






exp q1113957k(t)r(t) + qp 1113946t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897

exp1113882 minusqp

1113954be


qp

1113954b1113946

t

0e


qp

1113954br0

+qp

1113954b1113946

t

0[θ(s) + bξ(s)]ds1113883

times exp 1113946t

0

b


qp

1113954be

b1113954(sminus T)b1113890 1113891dW1(s)1113896 1113897

(A19)



11138681113868111386811138681113868111386811138681113868q

1113957D3(t) 1113957M(t) (A20)


r(τn) I(τn))1113966 1113967nisinN


Data Availability






Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018









20(t)x

2+ a(t)Vr




111388212Vxxσ

21(t)x

2π21 + 1113876Vxσ1(t)η(t)x


+ Vxrbσ1(t)x

+ VxIσ0(t)σ1(t)ρx1113877π11113883

+ supπ2isinR

12Vxxσ

22(t)x

2π22 + Vxλ(t)σ2(t)xπ21113882 1113883

+ supuisin[0infin)

12Vxxσ

23(t)u

2+ c(t)Vxu1113882 1113883

(18)

V(T x I r) 1p

xp (19)



p (20)




σ1(t)

Gx

xGxx

minusσ0(t)ρσ1(t)

GIx

xGxx

minusb

σ1(t)

Gxr

xGxx

+σ0(t)ρσ1(t)

(21)


xGxx

λ(t)

σ2(t) (22)

ulowast(t) minus

Gx

Gxx

c(t)

σ23(t) (23)


Gt gt

xp

p

Gx gxpminus 1


Gr gr

xp

p

Grr grr

xp

p

GI gI

xp

p

GII gII

xp

p

Gxr grxpminus 1

GxI gIxpminus 1

GrI grI

xp

p

(24)


0 gt

p+ g r + σ20 minus I1113872 1113873 +


1p

+12b2grr

p

+gI

pβ(α minus I) +

12σ20

gII


gIr

pbρσ0

minusg

p minus 1η2

2minus12

g(p minus 1)σ20ρ2

minusg2

I

g(p minus 1)

σ20ρ2

2

minusg2

r

g(p minus 1)

b2

2+ gησ0ρ minus

gI


gr

p minus 1bη


minusgrgI


g

p minus 1λ2

2

minusg

p minus 1c2

2σ23

(25)



(26)



gt fprimeeΔ + feΔ


gr fkeΔ

gI fzeΔ

gIr fkzeΔ

grr fk2eΔ

gII fz2eΔ

(27)


0 kprimep

+ 11113890 1113891fr +zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ 1113882σ20 +12


2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz minus

λ2

2(p minus 1)

minusc2

2σ23(p minus 1)1113883f +

ak

pf

(28)



h(t) ≔ σ20 +12

(p minus 1)σ20 +(1113957a + bξ)k

p+

b2k2

2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz

minusλ2

2(p minus 1)minus

c2

2σ23(p minus 1)

(29)


kprime(t) minus p

k(T) 0

⎧⎨

⎩ (30)


z(T) 01113896 (31)


f(T) 11113896 (32)


0 kprimep

+ 11113890 1113891fr +zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ h(t)f (33)



k(t) p(T minus t)


t

0β(s)ds

1113946

T

t

eminus 1113938

s

0β(v)dvds


T

th(s)ds

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)



T

th(s)ds + k(t)r + z(t)I1113896 1113897x

p

(35)



σ1(t)

1p minus 1

minusb

σ1(t)

p


ρσ0(t)

σ1(t)

minusρσ0(t)

σ1(t)

p

p minus 1e1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds


σ2(t)

1p minus 1

ulowast(t) minus

c(t)

σ23(t)

1p minus 1

x

(36)



1113957h(t) ≔ σ20 +12


p+

b2k2

2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz minus

λ2

2(p minus 1)

minusc2

2σ23(p minus 1)

(37)



1113957k(T) 0

⎧⎨

⎩ (38)


1113957z(T) 01113896 (39)

1113957fprime(t) + p1113957h(t)1113957f(t) 0

1113957f(T) 11113896 (40)


0 kprimep

minus1113954b

pk + 11113890 1113891fr +

zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ 1113957hf (41)



1113957k(t) p


1113957z(t) minus pe1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds

1113957f(t) eminus p 1113938

T

t

~h(s)ds

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)



T

t

1113957h(s)ds + 1113957k(t)r + 1113957z(t)I1113896 1113897xp

(43)



σ1(t)

1p minus 1

minus1

σ1(t)

p

p minus 11 minus e

b(tminus T)1113960 1113961

minusρσ0(t)

σ1(t)

p

p minus 1e1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds +

ρσ0(t)

σ1(t)


σ2(t)

1p minus 1


σ23(t)

1p minus 1

x

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)











I(τn) r(τn))1113966 1113967nisinN is






EtxrI

1p

Xπ1113957T1113874 1113875

p

1113890 1113891leG(t x r I) (45)


V(t x r I) EtxrI

1p

Xπ1113957lowastT1113874 1113875

p

1113890 1113891 G(t x r I) (46)


EtxrI

1p

Xπ1113957T1113874 1113875

p

1113890 1113891leG(t x r I) (47)


V(t x r I) EtxrI

1p

Xπ1113957lowastT1113874 1113875

p

1113890 1113891 G(t x r I) (48)







p = 04p = 05p = 06

0 10 20 30 40 50 60 70 80minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0


0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

p = 04p = 05p = 06




6 Conclusion


Appendix



dXlowastt X



σ23(t)

11 minus p

1113877dt +c(t)

σ3(t)

11 minus p

dW3(t)



(A1)



t


t

0I(s)ds

+1

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + 1113946

t


+ 1113946t


t

0σ0(s)dW0(s)1113883

(A2)

where

D1(t) X0 exp1113946t


+ σ20(s) +c2(s)


c2(s)



2


2

2minusσ20(s)

2


(A3)




any fixed qgt 1

0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0




supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q


fixed qgt 1


11138681113868111386811138681113868111386811138681113868q

1


T


lowastt( 1113857

pq

D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qk(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A5)

where we set


T

th(s)ds1113896 1113897D

p1(t) (A6)


0 sdr(s) + 1113938t



0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



+ qp 1113946t


(A7)


z(t)I(t) z(0)I0 + 1113946t


t

0z(s)dI(s)

z(0)I0 + 1113946t


t

0z(s) β(s)[α(s)1113864


z(0)I0 + 1113946t

0pI(s)ds + 1113946

t

0z(s)β(s)α(s)ds

+ 1113946t

0σ0(s)z(s)dW0(s)

(A8)




t

0σ0(s)dW0(s)1113896 1113897

exp1113882qz(0)I0 + q 1113946t

0β(s)α(s)z(s)ds

+ q 1113946t


t

0σ0(s)dW0(s)1113883

(A9)



11138681113868111386811138681113868111386811138681113868q

D2(t)1113868111386811138681113868

1113868111386811138681113868qexp1113882qpTr0 + qz(0)I0 + qp 1113946

t

0a(s)(T minus s)ds

+ q 1113946t

0β(s)α(s)z(s)ds1113883

middot exp1113882qp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t




+ q 1113946t


(A10)


Let

M(t) exp minus12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds minus

12q2p2

1113946t


2ds1113896

minus12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds minus12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

minus q2pρ1113946

t


middot expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t


+ qp 1113946t


t


D3(t) D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qpTr0 + qz(0)I0 + qp 1113946

t


t

0β(s)α(s)z(s)ds1113896 1113897

middot exp12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds +

12q2p2

1113946t


2ds1113896

+12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds +12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

+q2pρ1113946

t


(A11)



11138681113868111386811138681113868111386811138681113868q

D3(t)M(t) (A12)


E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875 E D3 τn( 1113857 middot M τn( 11138571113858 1113859

le suptisin[0T]


D3(t)ltinfin

(A13)


supnisinN

E G τn Xτn r τn( 1113857 I τn( 11138571113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A14)



supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A15)



11138681113868111386811138681113868111386811138681113868q

1


T

t


pq

1113957D2(t)1113868111386811138681113868

1113868111386811138681113868q exp q1113957k(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A16)


where we let

1113957D2(t) 1pexp minus p 1113946

T

t

1113957h(s)ds1113896 1113897Dp1(t) (A17)






exp q1113957k(t)r(t) + qp 1113946t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897

exp1113882 minusqp

1113954be


qp

1113954b1113946

t

0e


qp

1113954br0

+qp

1113954b1113946

t

0[θ(s) + bξ(s)]ds1113883

times exp 1113946t

0

b


qp

1113954be

b1113954(sminus T)b1113890 1113891dW1(s)1113896 1113897

(A19)



11138681113868111386811138681113868111386811138681113868q

1113957D3(t) 1113957M(t) (A20)


r(τn) I(τn))1113966 1113967nisinN


Data Availability






Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018








gt fprimeeΔ + feΔ


gr fkeΔ

gI fzeΔ

gIr fkzeΔ

grr fk2eΔ

gII fz2eΔ

(27)


0 kprimep

+ 11113890 1113891fr +zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ 1113882σ20 +12


2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz minus

λ2

2(p minus 1)

minusc2

2σ23(p minus 1)1113883f +

ak

pf

(28)



h(t) ≔ σ20 +12

(p minus 1)σ20 +(1113957a + bξ)k

p+

b2k2

2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz

minusλ2

2(p minus 1)minus

c2

2σ23(p minus 1)

(29)


kprime(t) minus p

k(T) 0

⎧⎨

⎩ (30)


z(T) 01113896 (31)


f(T) 11113896 (32)


0 kprimep

+ 11113890 1113891fr +zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ h(t)f (33)



k(t) p(T minus t)


t

0β(s)ds

1113946

T

t

eminus 1113938

s

0β(v)dvds


T

th(s)ds

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)



T

th(s)ds + k(t)r + z(t)I1113896 1113897x

p

(35)



σ1(t)

1p minus 1

minusb

σ1(t)

p


ρσ0(t)

σ1(t)

minusρσ0(t)

σ1(t)

p

p minus 1e1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds


σ2(t)

1p minus 1

ulowast(t) minus

c(t)

σ23(t)

1p minus 1

x

(36)



1113957h(t) ≔ σ20 +12


p+

b2k2

2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz minus

λ2

2(p minus 1)

minusc2

2σ23(p minus 1)

(37)



1113957k(T) 0

⎧⎨

⎩ (38)


1113957z(T) 01113896 (39)

1113957fprime(t) + p1113957h(t)1113957f(t) 0

1113957f(T) 11113896 (40)


0 kprimep

minus1113954b

pk + 11113890 1113891fr +

zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ 1113957hf (41)



1113957k(t) p


1113957z(t) minus pe1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds

1113957f(t) eminus p 1113938

T

t

~h(s)ds

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)



T

t

1113957h(s)ds + 1113957k(t)r + 1113957z(t)I1113896 1113897xp

(43)



σ1(t)

1p minus 1

minus1

σ1(t)

p

p minus 11 minus e

b(tminus T)1113960 1113961

minusρσ0(t)

σ1(t)

p

p minus 1e1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds +

ρσ0(t)

σ1(t)


σ2(t)

1p minus 1


σ23(t)

1p minus 1

x

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)











I(τn) r(τn))1113966 1113967nisinN is






EtxrI

1p

Xπ1113957T1113874 1113875

p

1113890 1113891leG(t x r I) (45)


V(t x r I) EtxrI

1p

Xπ1113957lowastT1113874 1113875

p

1113890 1113891 G(t x r I) (46)


EtxrI

1p

Xπ1113957T1113874 1113875

p

1113890 1113891leG(t x r I) (47)


V(t x r I) EtxrI

1p

Xπ1113957lowastT1113874 1113875

p

1113890 1113891 G(t x r I) (48)







p = 04p = 05p = 06

0 10 20 30 40 50 60 70 80minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0


0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

p = 04p = 05p = 06




6 Conclusion


Appendix



dXlowastt X



σ23(t)

11 minus p

1113877dt +c(t)

σ3(t)

11 minus p

dW3(t)



(A1)



t


t

0I(s)ds

+1

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + 1113946

t


+ 1113946t


t

0σ0(s)dW0(s)1113883

(A2)

where

D1(t) X0 exp1113946t


+ σ20(s) +c2(s)


c2(s)



2


2

2minusσ20(s)

2


(A3)




any fixed qgt 1

0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0




supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q


fixed qgt 1


11138681113868111386811138681113868111386811138681113868q

1


T


lowastt( 1113857

pq

D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qk(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A5)

where we set


T

th(s)ds1113896 1113897D

p1(t) (A6)


0 sdr(s) + 1113938t



0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



+ qp 1113946t


(A7)


z(t)I(t) z(0)I0 + 1113946t


t

0z(s)dI(s)

z(0)I0 + 1113946t


t

0z(s) β(s)[α(s)1113864


z(0)I0 + 1113946t

0pI(s)ds + 1113946

t

0z(s)β(s)α(s)ds

+ 1113946t

0σ0(s)z(s)dW0(s)

(A8)




t

0σ0(s)dW0(s)1113896 1113897

exp1113882qz(0)I0 + q 1113946t

0β(s)α(s)z(s)ds

+ q 1113946t


t

0σ0(s)dW0(s)1113883

(A9)



11138681113868111386811138681113868111386811138681113868q

D2(t)1113868111386811138681113868

1113868111386811138681113868qexp1113882qpTr0 + qz(0)I0 + qp 1113946

t

0a(s)(T minus s)ds

+ q 1113946t

0β(s)α(s)z(s)ds1113883

middot exp1113882qp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t




+ q 1113946t


(A10)


Let

M(t) exp minus12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds minus

12q2p2

1113946t


2ds1113896

minus12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds minus12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

minus q2pρ1113946

t


middot expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t


+ qp 1113946t


t


D3(t) D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qpTr0 + qz(0)I0 + qp 1113946

t


t

0β(s)α(s)z(s)ds1113896 1113897

middot exp12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds +

12q2p2

1113946t


2ds1113896

+12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds +12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

+q2pρ1113946

t


(A11)



11138681113868111386811138681113868111386811138681113868q

D3(t)M(t) (A12)


E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875 E D3 τn( 1113857 middot M τn( 11138571113858 1113859

le suptisin[0T]


D3(t)ltinfin

(A13)


supnisinN

E G τn Xτn r τn( 1113857 I τn( 11138571113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A14)



supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A15)



11138681113868111386811138681113868111386811138681113868q

1


T

t


pq

1113957D2(t)1113868111386811138681113868

1113868111386811138681113868q exp q1113957k(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A16)


where we let

1113957D2(t) 1pexp minus p 1113946

T

t

1113957h(s)ds1113896 1113897Dp1(t) (A17)






exp q1113957k(t)r(t) + qp 1113946t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897

exp1113882 minusqp

1113954be


qp

1113954b1113946

t

0e


qp

1113954br0

+qp

1113954b1113946

t

0[θ(s) + bξ(s)]ds1113883

times exp 1113946t

0

b


qp

1113954be

b1113954(sminus T)b1113890 1113891dW1(s)1113896 1113897

(A19)



11138681113868111386811138681113868111386811138681113868q

1113957D3(t) 1113957M(t) (A20)


r(τn) I(τn))1113966 1113967nisinN


Data Availability






Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018









1113957h(t) ≔ σ20 +12


p+

b2k2

2p+αβp

z +σ20z

2

2p


pkz minus

η2

2(p minus 1)minus

p minus 12

ρ2σ20

minusρ2σ20

2(p minus 1)z2

minusb2

2(p minus 1)k2


z

minusηb


2z + σ0bρk minus

σ0ρb

p minus 1kz minus

λ2

2(p minus 1)

minusc2

2σ23(p minus 1)

(37)



1113957k(T) 0

⎧⎨

⎩ (38)


1113957z(T) 01113896 (39)

1113957fprime(t) + p1113957h(t)1113957f(t) 0

1113957f(T) 11113896 (40)


0 kprimep

minus1113954b

pk + 11113890 1113891fr +

zprimep

minusβz

pminus 11113890 1113891fI +

fprimep

+ 1113957hf (41)



1113957k(t) p


1113957z(t) minus pe1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds

1113957f(t) eminus p 1113938

T

t

~h(s)ds

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)



T

t

1113957h(s)ds + 1113957k(t)r + 1113957z(t)I1113896 1113897xp

(43)



σ1(t)

1p minus 1

minus1

σ1(t)

p

p minus 11 minus e

b(tminus T)1113960 1113961

minusρσ0(t)

σ1(t)

p

p minus 1e1113938

t

0β(s)ds

1113946T

te

minus 1113938s

0β(v)dvds +

ρσ0(t)

σ1(t)


σ2(t)

1p minus 1


σ23(t)

1p minus 1

x

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)











I(τn) r(τn))1113966 1113967nisinN is






EtxrI

1p

Xπ1113957T1113874 1113875

p

1113890 1113891leG(t x r I) (45)


V(t x r I) EtxrI

1p

Xπ1113957lowastT1113874 1113875

p

1113890 1113891 G(t x r I) (46)


EtxrI

1p

Xπ1113957T1113874 1113875

p

1113890 1113891leG(t x r I) (47)


V(t x r I) EtxrI

1p

Xπ1113957lowastT1113874 1113875

p

1113890 1113891 G(t x r I) (48)







p = 04p = 05p = 06

0 10 20 30 40 50 60 70 80minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0


0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

p = 04p = 05p = 06




6 Conclusion


Appendix



dXlowastt X



σ23(t)

11 minus p

1113877dt +c(t)

σ3(t)

11 minus p

dW3(t)



(A1)



t


t

0I(s)ds

+1

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + 1113946

t


+ 1113946t


t

0σ0(s)dW0(s)1113883

(A2)

where

D1(t) X0 exp1113946t


+ σ20(s) +c2(s)


c2(s)



2


2

2minusσ20(s)

2


(A3)




any fixed qgt 1

0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0




supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q


fixed qgt 1


11138681113868111386811138681113868111386811138681113868q

1


T


lowastt( 1113857

pq

D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qk(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A5)

where we set


T

th(s)ds1113896 1113897D

p1(t) (A6)


0 sdr(s) + 1113938t



0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



+ qp 1113946t


(A7)


z(t)I(t) z(0)I0 + 1113946t


t

0z(s)dI(s)

z(0)I0 + 1113946t


t

0z(s) β(s)[α(s)1113864


z(0)I0 + 1113946t

0pI(s)ds + 1113946

t

0z(s)β(s)α(s)ds

+ 1113946t

0σ0(s)z(s)dW0(s)

(A8)




t

0σ0(s)dW0(s)1113896 1113897

exp1113882qz(0)I0 + q 1113946t

0β(s)α(s)z(s)ds

+ q 1113946t


t

0σ0(s)dW0(s)1113883

(A9)



11138681113868111386811138681113868111386811138681113868q

D2(t)1113868111386811138681113868

1113868111386811138681113868qexp1113882qpTr0 + qz(0)I0 + qp 1113946

t

0a(s)(T minus s)ds

+ q 1113946t

0β(s)α(s)z(s)ds1113883

middot exp1113882qp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t




+ q 1113946t


(A10)


Let

M(t) exp minus12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds minus

12q2p2

1113946t


2ds1113896

minus12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds minus12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

minus q2pρ1113946

t


middot expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t


+ qp 1113946t


t


D3(t) D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qpTr0 + qz(0)I0 + qp 1113946

t


t

0β(s)α(s)z(s)ds1113896 1113897

middot exp12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds +

12q2p2

1113946t


2ds1113896

+12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds +12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

+q2pρ1113946

t


(A11)



11138681113868111386811138681113868111386811138681113868q

D3(t)M(t) (A12)


E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875 E D3 τn( 1113857 middot M τn( 11138571113858 1113859

le suptisin[0T]


D3(t)ltinfin

(A13)


supnisinN

E G τn Xτn r τn( 1113857 I τn( 11138571113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A14)



supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A15)



11138681113868111386811138681113868111386811138681113868q

1


T

t


pq

1113957D2(t)1113868111386811138681113868

1113868111386811138681113868q exp q1113957k(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A16)


where we let

1113957D2(t) 1pexp minus p 1113946

T

t

1113957h(s)ds1113896 1113897Dp1(t) (A17)






exp q1113957k(t)r(t) + qp 1113946t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897

exp1113882 minusqp

1113954be


qp

1113954b1113946

t

0e


qp

1113954br0

+qp

1113954b1113946

t

0[θ(s) + bξ(s)]ds1113883

times exp 1113946t

0

b


qp

1113954be

b1113954(sminus T)b1113890 1113891dW1(s)1113896 1113897

(A19)



11138681113868111386811138681113868111386811138681113868q

1113957D3(t) 1113957M(t) (A20)


r(τn) I(τn))1113966 1113967nisinN


Data Availability






Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018











EtxrI

1p

Xπ1113957T1113874 1113875

p

1113890 1113891leG(t x r I) (45)


V(t x r I) EtxrI

1p

Xπ1113957lowastT1113874 1113875

p

1113890 1113891 G(t x r I) (46)


EtxrI

1p

Xπ1113957T1113874 1113875

p

1113890 1113891leG(t x r I) (47)


V(t x r I) EtxrI

1p

Xπ1113957lowastT1113874 1113875

p

1113890 1113891 G(t x r I) (48)







p = 04p = 05p = 06

0 10 20 30 40 50 60 70 80minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0


0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

p = 04p = 05p = 06




6 Conclusion


Appendix



dXlowastt X



σ23(t)

11 minus p

1113877dt +c(t)

σ3(t)

11 minus p

dW3(t)



(A1)



t


t

0I(s)ds

+1

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + 1113946

t


+ 1113946t


t

0σ0(s)dW0(s)1113883

(A2)

where

D1(t) X0 exp1113946t


+ σ20(s) +c2(s)


c2(s)



2


2

2minusσ20(s)

2


(A3)




any fixed qgt 1

0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0




supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q


fixed qgt 1


11138681113868111386811138681113868111386811138681113868q

1


T


lowastt( 1113857

pq

D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qk(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A5)

where we set


T

th(s)ds1113896 1113897D

p1(t) (A6)


0 sdr(s) + 1113938t



0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



+ qp 1113946t


(A7)


z(t)I(t) z(0)I0 + 1113946t


t

0z(s)dI(s)

z(0)I0 + 1113946t


t

0z(s) β(s)[α(s)1113864


z(0)I0 + 1113946t

0pI(s)ds + 1113946

t

0z(s)β(s)α(s)ds

+ 1113946t

0σ0(s)z(s)dW0(s)

(A8)




t

0σ0(s)dW0(s)1113896 1113897

exp1113882qz(0)I0 + q 1113946t

0β(s)α(s)z(s)ds

+ q 1113946t


t

0σ0(s)dW0(s)1113883

(A9)



11138681113868111386811138681113868111386811138681113868q

D2(t)1113868111386811138681113868

1113868111386811138681113868qexp1113882qpTr0 + qz(0)I0 + qp 1113946

t

0a(s)(T minus s)ds

+ q 1113946t

0β(s)α(s)z(s)ds1113883

middot exp1113882qp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t




+ q 1113946t


(A10)


Let

M(t) exp minus12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds minus

12q2p2

1113946t


2ds1113896

minus12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds minus12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

minus q2pρ1113946

t


middot expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t


+ qp 1113946t


t


D3(t) D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qpTr0 + qz(0)I0 + qp 1113946

t


t

0β(s)α(s)z(s)ds1113896 1113897

middot exp12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds +

12q2p2

1113946t


2ds1113896

+12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds +12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

+q2pρ1113946

t


(A11)



11138681113868111386811138681113868111386811138681113868q

D3(t)M(t) (A12)


E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875 E D3 τn( 1113857 middot M τn( 11138571113858 1113859

le suptisin[0T]


D3(t)ltinfin

(A13)


supnisinN

E G τn Xτn r τn( 1113857 I τn( 11138571113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A14)



supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A15)



11138681113868111386811138681113868111386811138681113868q

1


T

t


pq

1113957D2(t)1113868111386811138681113868

1113868111386811138681113868q exp q1113957k(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A16)


where we let

1113957D2(t) 1pexp minus p 1113946

T

t

1113957h(s)ds1113896 1113897Dp1(t) (A17)






exp q1113957k(t)r(t) + qp 1113946t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897

exp1113882 minusqp

1113954be


qp

1113954b1113946

t

0e


qp

1113954br0

+qp

1113954b1113946

t

0[θ(s) + bξ(s)]ds1113883

times exp 1113946t

0

b


qp

1113954be

b1113954(sminus T)b1113890 1113891dW1(s)1113896 1113897

(A19)



11138681113868111386811138681113868111386811138681113868q

1113957D3(t) 1113957M(t) (A20)


r(τn) I(τn))1113966 1113967nisinN


Data Availability






Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018









6 Conclusion


Appendix



dXlowastt X



σ23(t)

11 minus p

1113877dt +c(t)

σ3(t)

11 minus p

dW3(t)



(A1)



t


t

0I(s)ds

+1

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + 1113946

t


+ 1113946t


t

0σ0(s)dW0(s)1113883

(A2)

where

D1(t) X0 exp1113946t


+ σ20(s) +c2(s)


c2(s)



2


2

2minusσ20(s)

2


(A3)




any fixed qgt 1

0 10 20 30 40 50 60 70 80ndash14

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0




supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q


fixed qgt 1


11138681113868111386811138681113868111386811138681113868q

1


T


lowastt( 1113857

pq

D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qk(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A5)

where we set


T

th(s)ds1113896 1113897D

p1(t) (A6)


0 sdr(s) + 1113938t



0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



+ qp 1113946t


(A7)


z(t)I(t) z(0)I0 + 1113946t


t

0z(s)dI(s)

z(0)I0 + 1113946t


t

0z(s) β(s)[α(s)1113864


z(0)I0 + 1113946t

0pI(s)ds + 1113946

t

0z(s)β(s)α(s)ds

+ 1113946t

0σ0(s)z(s)dW0(s)

(A8)




t

0σ0(s)dW0(s)1113896 1113897

exp1113882qz(0)I0 + q 1113946t

0β(s)α(s)z(s)ds

+ q 1113946t


t

0σ0(s)dW0(s)1113883

(A9)



11138681113868111386811138681113868111386811138681113868q

D2(t)1113868111386811138681113868

1113868111386811138681113868qexp1113882qpTr0 + qz(0)I0 + qp 1113946

t

0a(s)(T minus s)ds

+ q 1113946t

0β(s)α(s)z(s)ds1113883

middot exp1113882qp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t




+ q 1113946t


(A10)


Let

M(t) exp minus12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds minus

12q2p2

1113946t


2ds1113896

minus12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds minus12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

minus q2pρ1113946

t


middot expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t


+ qp 1113946t


t


D3(t) D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qpTr0 + qz(0)I0 + qp 1113946

t


t

0β(s)α(s)z(s)ds1113896 1113897

middot exp12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds +

12q2p2

1113946t


2ds1113896

+12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds +12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

+q2pρ1113946

t


(A11)



11138681113868111386811138681113868111386811138681113868q

D3(t)M(t) (A12)


E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875 E D3 τn( 1113857 middot M τn( 11138571113858 1113859

le suptisin[0T]


D3(t)ltinfin

(A13)


supnisinN

E G τn Xτn r τn( 1113857 I τn( 11138571113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A14)



supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A15)



11138681113868111386811138681113868111386811138681113868q

1


T

t


pq

1113957D2(t)1113868111386811138681113868

1113868111386811138681113868q exp q1113957k(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A16)


where we let

1113957D2(t) 1pexp minus p 1113946

T

t

1113957h(s)ds1113896 1113897Dp1(t) (A17)






exp q1113957k(t)r(t) + qp 1113946t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897

exp1113882 minusqp

1113954be


qp

1113954b1113946

t

0e


qp

1113954br0

+qp

1113954b1113946

t

0[θ(s) + bξ(s)]ds1113883

times exp 1113946t

0

b


qp

1113954be

b1113954(sminus T)b1113890 1113891dW1(s)1113896 1113897

(A19)



11138681113868111386811138681113868111386811138681113868q

1113957D3(t) 1113957M(t) (A20)


r(τn) I(τn))1113966 1113967nisinN


Data Availability






Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018








supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q


fixed qgt 1


11138681113868111386811138681113868111386811138681113868q

1


T


lowastt( 1113857

pq

D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qk(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A5)

where we set


T

th(s)ds1113896 1113897D

p1(t) (A6)


0 sdr(s) + 1113938t



0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



+ qp 1113946t


(A7)


z(t)I(t) z(0)I0 + 1113946t


t

0z(s)dI(s)

z(0)I0 + 1113946t


t

0z(s) β(s)[α(s)1113864


z(0)I0 + 1113946t

0pI(s)ds + 1113946

t

0z(s)β(s)α(s)ds

+ 1113946t

0σ0(s)z(s)dW0(s)

(A8)




t

0σ0(s)dW0(s)1113896 1113897

exp1113882qz(0)I0 + q 1113946t

0β(s)α(s)z(s)ds

+ q 1113946t


t

0σ0(s)dW0(s)1113883

(A9)



11138681113868111386811138681113868111386811138681113868q

D2(t)1113868111386811138681113868

1113868111386811138681113868qexp1113882qpTr0 + qz(0)I0 + qp 1113946

t

0a(s)(T minus s)ds

+ q 1113946t

0β(s)α(s)z(s)ds1113883

middot exp1113882qp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t




+ q 1113946t


(A10)


Let

M(t) exp minus12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds minus

12q2p2

1113946t


2ds1113896

minus12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds minus12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

minus q2pρ1113946

t


middot expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t


+ qp 1113946t


t


D3(t) D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qpTr0 + qz(0)I0 + qp 1113946

t


t

0β(s)α(s)z(s)ds1113896 1113897

middot exp12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds +

12q2p2

1113946t


2ds1113896

+12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds +12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

+q2pρ1113946

t


(A11)



11138681113868111386811138681113868111386811138681113868q

D3(t)M(t) (A12)


E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875 E D3 τn( 1113857 middot M τn( 11138571113858 1113859

le suptisin[0T]


D3(t)ltinfin

(A13)


supnisinN

E G τn Xτn r τn( 1113857 I τn( 11138571113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A14)



supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A15)



11138681113868111386811138681113868111386811138681113868q

1


T

t


pq

1113957D2(t)1113868111386811138681113868

1113868111386811138681113868q exp q1113957k(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A16)


where we let

1113957D2(t) 1pexp minus p 1113946

T

t

1113957h(s)ds1113896 1113897Dp1(t) (A17)






exp q1113957k(t)r(t) + qp 1113946t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897

exp1113882 minusqp

1113954be


qp

1113954b1113946

t

0e


qp

1113954br0

+qp

1113954b1113946

t

0[θ(s) + bξ(s)]ds1113883

times exp 1113946t

0

b


qp

1113954be

b1113954(sminus T)b1113890 1113891dW1(s)1113896 1113897

(A19)



11138681113868111386811138681113868111386811138681113868q

1113957D3(t) 1113957M(t) (A20)


r(τn) I(τn))1113966 1113967nisinN


Data Availability






Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018








Let

M(t) exp minus12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds minus

12q2p2

1113946t


2ds1113896

minus12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds minus12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

minus q2pρ1113946

t


middot expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s) + qp 1113946

t


+ qp 1113946t


t


D3(t) D2(t)1113868111386811138681113868

1113868111386811138681113868q exp qpTr0 + qz(0)I0 + qp 1113946

t


t

0β(s)α(s)z(s)ds1113896 1113897

middot exp12

q2p2

(1 minus p)21113946

t

0

c2(s)

σ23(s)ds +

12q2p2

1113946t


2ds1113896

+12q2p2

1113946t

0σ22(s) πlowast2(s)( 1113857

2ds +12q2

1113946t

0σ0(s)z(s) minus pσ0(s)1113858 1113859

2ds

+q2pρ1113946

t


(A11)



11138681113868111386811138681113868111386811138681113868q

D3(t)M(t) (A12)


E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875 E D3 τn( 1113857 middot M τn( 11138571113858 1113859

le suptisin[0T]


D3(t)ltinfin

(A13)


supnisinN

E G τn Xτn r τn( 1113857 I τn( 11138571113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A14)



supnisinN

E G τn Xlowastτn

r τn( 1113857 I τn( 11138571113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868q

1113874 1113875ltinfin (A15)



11138681113868111386811138681113868111386811138681113868q

1


T

t


pq

1113957D2(t)1113868111386811138681113868

1113868111386811138681113868q exp q1113957k(t)r(t) + qp 1113946

t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897



t

0σ0(s)dW0(s)1113896 1113897

times expqp

1 minus p1113946

t

0

c(s)

σ3(s)dW3(s)1113896 1113897exp qp 1113946

t

0σ2(s)πlowast2(s)dW2(s)1113896 1113897

(A16)


where we let

1113957D2(t) 1pexp minus p 1113946

T

t

1113957h(s)ds1113896 1113897Dp1(t) (A17)






exp q1113957k(t)r(t) + qp 1113946t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897

exp1113882 minusqp

1113954be


qp

1113954b1113946

t

0e


qp

1113954br0

+qp

1113954b1113946

t

0[θ(s) + bξ(s)]ds1113883

times exp 1113946t

0

b


qp

1113954be

b1113954(sminus T)b1113890 1113891dW1(s)1113896 1113897

(A19)



11138681113868111386811138681113868111386811138681113868q

1113957D3(t) 1113957M(t) (A20)


r(τn) I(τn))1113966 1113967nisinN


Data Availability






Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018








where we let

1113957D2(t) 1pexp minus p 1113946

T

t

1113957h(s)ds1113896 1113897Dp1(t) (A17)






exp q1113957k(t)r(t) + qp 1113946t

0r(s)ds + qp 1113946

t

0σ1(s)πlowast1(s)dW1(s)1113896 1113897

exp1113882 minusqp

1113954be


qp

1113954b1113946

t

0e


qp

1113954br0

+qp

1113954b1113946

t

0[θ(s) + bξ(s)]ds1113883

times exp 1113946t

0

b


qp

1113954be

b1113954(sminus T)b1113890 1113891dW1(s)1113896 1113897

(A19)



11138681113868111386811138681113868111386811138681113868q

1113957D3(t) 1113957M(t) (A20)


r(τn) I(τn))1113966 1113967nisinN


Data Availability






Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018









































Journal of












Journal of







Volume 2018






Volume 2018








optimal investment-reinsurance policy with stochastic...

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