ruin theory in risk models based on time series for count random … · 2017-09-28 · ruin theory...

Ruin Theory in risk models based on time series forcount random variables

Etienne Marceau, Ph.D. A.S.A.

with H. Cossette, Florent Toureille

A celebration of Hans Gerber�s contributions (Waterloo, Canada)

Université Laval

June 18, 2013

Étienne Marceau (Université Laval) Poisson frais June 18, 2013 1 / 54

1. Introduction

The objective of the talk is examine an extension of the classicaldiscrete-time risk model

Our extension is loosely inspired on Gerber�s own extension of thismodel

Gerber�s extension is based on gaussian linear time series

Our extension is based on time series for count data

We examine the computation of the following ruin measures :

Adjustment coe¢ cientGerber-Shiu fonction on �nite-time horizonJoint probabilities of time of ruin, de�cit at ruin and surplus before ruinDynamic VaR and TVaR

In this talk, we "celebrate" among numerous Gerber�s contributions

Gerber�s gaussian linear risk modelSo-called Gerber-Shiu functionComputation of various ruin measures


2. Classic discrete-time risk model

The classical discrete time risk model is due to De Finetti (1957) (seealso e.g. Bühlmann (1970), Gerber (1979) and Dickson (2005)).

We consider an insurance portfolio

Aggregate claim amount in period k 2 N+: r.v. Wk

0j0��| {z }W1

j1��| {z }W2

j2��| {z }W3

j3��| {z }W4

j4��| {z }W5

j5�

W = fWk , k 2 N+g = sequence of i.i.d. r.v.�sPremium income per period: π = (1+ η)E [W ]

Strictly positive security margin: η > 0



Surplus process: U = fUk , k 2 Ngfor k = 0, U0 = u = initial surplusfor k 2 N+, Uk = Uk�1 + π �Wk = u �∑kj=1

�Wj � π

�Time of ruin: r.v. τ with

τ =

(inf

k2f1,2,3,...gfk,Uk < 0g , if Uk goes below 0 at least once

∞, if Uk never goes below 0

Finite-time ruin probablity over n periods : ψ (u, n) = Pr (τ � n)In�nite-time ruin probability : ψ (u) = lim

n!∞ψ (u, n)



Net loss in period j 2 N+: Lj = (Wj � π)

L1, L2, ... i.i.d. with E [Lj ] = E [Lj ]� π < 0 (since η > 0)

Random walk with negative drift: Y = fYk , k 2 Ng: Y0 = 0 and

Yk =k

∑j=1Lj

Maximum net cumulative loss process: Z = fZk , k 2 Ng

Zk = maxj=0,1,2,..,k

fYjg

Finite-time ruin probablity over n periods:

ψ (u, n) = Pr (τ � n) = Pr (Zn > u) = 1� FZn (u)



The distribution of Zn has a mass at 0

Risk management: the knowledge of FZn can be used to determinethe initial capital

We consider 2 risk measures based on ZnDynamic VaRDynamic TVaR

Dynamic VaR :VaRκ (Zn) = F�1Zn (κ)

(see e.g. Albrecher (2009), Tru¢ n et al. (2010), Cossette et al.(2013)).

Interpretation: VaRκ (Zn) = capital required such that the probabilitythat the maximum Zn of the random walk Y over the �rst n periodsexceeds that capital is equal to 1� κ.



Dynamic TVaR :

TVaRκ (Zn) =1

1� κ

Z ∞

1�κVaR (d )v (Zn) dv

equals to

EhZn � 1fZn>VaRκ(Zn)g

i+ VaRu (Zn) (FZn (VaRκ (Zn))� κ)

1� κ

(Cossette et al. (2013))

Compared to the dynamic VaR, the dynamic TVaR has the advantageof being more sensitive to the stochastic behavior of Zn in the tail ofits distribution.


3. Gerber�s linear gaussian (discrete time) risk model

Some papers consider various models with temporal dependence.

Gerber (1982) proposes extension :

Title : Ruin Theory in the linear model, IME 1982The risk model is based on gaussian linear time series (e.g. AR,ARMA, etc.)Gerber examines the approximation of ruin probabilities

Variants and extension

Promislow (1991) : upper bounds for ψ (u) in a similar risk modelChrist and Steinebach (1995) : empirical mgf type estimator of theadjustment coe¢ cient in Gerber�s risk model.Yang and Zhang (2003) : both exponential and non-exponential upperbounds for ψ (u) in an extension to Gerber�s risk model.Tru�n et al (2010) : interesting application of the Gerber�s risk modelin the context of insurance cycles.


3. Gerber�s extension : Gerber�s linear gaussian risk model

As stated in almost all actuarial textbooks, compound distributionsare the corner stones of several risk models in risk theory.

The linear risk models such as the Gaussian AR(1) and ARMA(p, q)do not lead to compound distributions for the rvs W .

We adapt the idea of Gerber

We assume that Wk follows a compound distribution

Wk =

(∑Nkj=1 Bk ,j , Nk > 0

0, Nk = 0

r.v. Nk : number of claims in period k�Bk ,j , j 2 N+

= individual claim amounts in period k (positive r.v.�s)�

Bk ,j , j 2 N+= sequence of i.i.d. r.v�s and also independent of Nk

Premium income per period: π = (1+ η)E [W ]

Strictly positive security margin: η > 0


4. Ruin theory based on time series for count data

We consider risk models based on compound distributions assumingtime series models for count data for N = fNk , k 2 N+g.4 categories of time series models for count data for N.

Models based on thinning :

N = Poisson INAR, Poisson INMA, Poisson INARMA, etc.These models are based on appropriate thinning operations whichreplace the scalar multiplications by a fraction in the Gaussian ARMAtime series(see e.g. Mckenzie (1986, 1988, 2003), and Joe (1997))Quddus (2008) and Gourieroux and Jasiak (2004) : car accident countdataFreeland (1998) and Freeland and McCabe (2004) : Worker�sCompensation Board of British ColumbiaKremer (1995) adapts the theory of INAR processes to the context ofIBNR-predictions


4. Ruin theory based on count ...

Other types of models of times series for count data :

Models based on Markov chains. N = Markov chain of order 1 or more.Models based on a speci�c conditional distribution with stochasticparameters.

N has a dependence structure based on an underlying process ΘΘ = ARMA time series or hidden discrete time Markov chain de�nedon a �nite time spaceMalyshkina, Mannering and Tarko (2009) explore two-state Markovswitching count data models to study accident frequencies

Models based on copulas : marginals are �xed and the dependencestructure of N is based on a copula (see e.g. Frees and Wang (2006))

Reviews on time series models for count data :

McKenzie (2003)Cameron and Trivedi (1998)Kedem and Fokianos (2002)

In this talk, we focus on the case N = Poisson INAR1 process


Risk Model �Poisson INAR(1)

Dynamic of N = fNk , k 2 N+g

Nk = α �Nk�1 + εk , (1)

for k = 2, 3, ...ε = fεk , k 2 N+g : sequence of i.i.d. r.v.�s withεk � Pois ((1� α)λ) where α 2 [0, 1].α �Nk�1 :

The operator "�" is used in models based on thinningWe have

α �Nk�1 =Nk�1

∑i=1

δk�1,i ,

where fδk�1,i , i = 1, 2, ...g is a sequence of i.i.d Bernoulli r.v�s withmean α and independent of Nk�1α = dependence parameter

Given (1), N is stationary with Nk � Pois (λ).Étienne Marceau (Université Laval) Poisson frais June 18, 2013 12 / 54


Interpretation in an insurance context :

nb of claimsin period k| {z }

Nk

=

nb of claimsfrom period k � 1leadind to claimsin period k| {z }α �Nk�1

+

nb of new claimsin period k| {z }

εk

Autocorrelation function for N

γN (h) = αh, h � 1,with γN (1) 2 [0, 1).Covariance between Wk and Wk+h :

Cov (Wk ,Wk+h) = λαhE [B ]2 ,

for h � 1.Étienne Marceau (Université Laval) Poisson frais June 18, 2013 13 / 54


Paths of N (α = 0)



Paths of N (α = 0.4)



Paths of N (α = 0.99)



Paths of V (α = 0)



Paths of V (α = 0.4)



Paths of V (α = 0.99)



Paths of Z (α = 0)



Paths of Z (α = 0.4)



Paths of Z (α = 0.99)


Adjustment coe¢ cient and asymptotic expression

De�ne the convex function

cn (r) =1nln�EherVn

i�=1nln�Eher (Sn�nπ)

i�=, (2)

where Sn = ∑nk=1Wk

Lundberg adjusment coe¢ cient : ρ is the solution to

c (r) = limn!∞

cn (r) = 0. (3)

(see e.g. Nyrhinen (1998), Müller and P�ug (2001))Asymptotic Lundberg-type result :

limu!∞

� ln (ψ (u))u

= ρ,

Adjustment coe¢ cient ρ : measure of dangerousness of an insuranceportfolio.The expression of cn (r) depends on the temporal dependencestructure for W .


Adjustment coe¢ cient �Poisson INAR(1)

Poisson AR(1). Assuming that αMB (r) < 1, the expression forc (r) is given by

c (r) =(1� α)2 λMB (r)1� (αMB (r))

� (1� α) λ� rπ.

Poisson AR(1). Impact of dependence parameter α.If α < α0, then ρ > ρ0.Dangerousness increases with the dependence parameter α.

Poisson AR(1) + Exponential claims. Assume that B � Exp (β)with mean 1

β . Then, we have

ρ =(1� α) βη

1+ η, (4)

where α 2 [0, 1). Special case : if α = 0, ρ = βη(1+η)

(independence).

(Cossette et al. (2010)).



Example.

λ = 0.4, B � Exp�

β = 12

�and η = 0.25 ; c = 1

approximation based on 5000 crude MC simulations, 1000 periods

α E [Z1000 ] Var (Z1000) VaR0.95 (Z1000) VaR0.99 (Z1000)0.2 9.646 151.904 35.018 55.4930.8 34.777 1964.363 127.714 191.145

approximation with asymptotic expression

α ρ � 1ρ ln (0.05) � 1ρ ln (0.01)0.2 0.08 37.447 57.5650.8 0.02 149.787 230.258



Example (suite)

Approximation of ψ (u) using importance sampling

α ρ eψ (0) eψ (10) eψ (20) eψ (30)0 0.1 0.771769 0.280084 0.103552 0.0378060.2 0.08 0.816092 0.364378 0.164038 0.0723330.4 0.06 0.850753 0.463960 0.255592 0.140231

ψ (u) " as α "Adaptation from e.g. Asmussen (2003).


Finite-time ruin probability

We present recursive formulas for the computation of �nite-time ruinmeasuresAdditional assumptions : π 2 N+, u 2 N, rv B is de�ned on N (tosimplify the notation)All formulas can be easily adapted to the case when the support isf0, 1h, 2h, ...g with h > 1Notation :

Pr (N1 = j) = pj , j 2 N

Pr (N2 = k2 jN1 = k1) = pk1,k2 , k1, k2 2 N

We compute ψ (u, n) with

ψ (u, n) =∞

∑k1=0

pk1ψ (u, njN1 = k1) .

whereψ (u, njN1 = k1) = Pr (τ � njN1 = k1) .



For n = 1, we have

ψ (u, 1jN1 = k1) =∞

∑l=u+c+1

f (l , k1),

where

f (j ; k1) =

8<:Pr (B1 + ...+ Bk1 = j) , if k1 > 01, if k1 = 0 and j = 00, if k1 = 0 and j 6= 0

.

For n � 2, we have

ψ (u, njN1 = k1) =∞

∑l=u+c+1

f (l , k1)

+∞

∑k2=0

pk1,k2u+c

∑j=0

f (j , k1)ψ (u + π � j , n� 1jN1 = k2) .



Example :

Poisson parameter λ = 0.4; claim amount : B � Geom�13

�, with

E [B ] = 2premium income : 1

α E [Z20 ] Var (Z20) VaR0.95 (Z20) TVaR0.95 (Z20)0 4.2668398 33.264562 12 17.7237380.2 4.6316362 40.952906 13 19.6544150.5 5.4468555 63.290655 16 24.3475080.8 7.0412564 135.708264 22 35.378478

E [Z20 ], Var (Z20) and TVaR0.95 (Z20) increase as α " (to be shown)Interpretation :

if the dependence relation is stronger, than the risk process is riskierthen the required amount as initial capital has to be higher


Finite-time Gerber-Shiu Function

Additional assumptions : π 2 N+, u 2 N, rv B is de�ned on N (tosimplify the notation)

In�nite-time GS function(1998) :

mδ(u) = E�v τw (Uτ� , jU(τ)j) 1fτ<∞gjU(0) = u

�w(x , y), for x , y � 0, is the penalty function at the time of ruin for thesurplus prior to ruin and the de�cit at ruin,v = e�δ < 1 is the discounted factor,1A is the indicator function, such that 1A = 1, if the event A occurs,and equals 0 otherwise.

Special cases

when w(x , y) = 1, for all x , y � 0, mδ(u) = Laplace Transform (LT)of τif δ = 0 and w(x , y) = 1 for all x , y 2 R, mδ(u) corresponds to ψ(u)



Extended version of the expected discounted penalty function over nperiods :

mδ(u, n) = E�v τw (Uτ� , jU(τ)j) 1fτ�ngjU(0) = u

�. (5)

(inspired by Morales (2010))

Special cases

when w(x , y) = 1 for all x , y � 0 and δ = 0, mδ(u, n) = ψ (u, n)

De�ne

mδ(u, n;N1 = k1) = E�v τw (Uτ� , jU(τ)j) 1fτ�ngjN1 = k1,U(0) = u

�.

We have

mδ(u, n) =∞

∑k1=0

pk1mδ(u, n;N1 = k1), (6)



Proposition. We have

mδ(u, n;N1 = k1) = v∞

∑k2=0

pk1,k2u+π

∑j=0

f (j ; k1)mδ(u+π� j , n� 1;N1 = k2)

+v∞

∑j=u+π+1

f (j ; k1) w(u, j � u � π)

where

f (j ; k1) =


. (7)

for n 2 N+ and for k1 2 N


Bivariate and trivariate probabilities

De�nitions inspired from Alfa and Drekic (2007) (for n 2 N+)

Bivariate joint probabilities for (τ,Uτ�) :

ζn,i (u) = Pr (τ = n,Uτ� = i jU(0) = u)

Bivariate joint probabilities for (τ, jUτ j) :

ξn,j (u) = Pr (τ = n, jUτ j = j jU(0) = u)

Trivariate joint probabilities for (τ,Uτ�, jUτ j) :

$n,i ,j (u) = Pr (τ = n,Uτ� = i , jUτ j = j jU(0) = u)

for i = 0, 1, 2, ...; j = 1, 2, ...


Bivariate joint probabilities for time at ruin and de�cit atruin

We de�ne

ξ(k1)n,j (u) = Pr (τ = n, jUτj = j jU(0) = u,N1 = k1)

for n 2 N+, u 2 N, j 2 N

We have the following relationship

ξn,j (u) =∞

∑k1=0

pk1ξ(k1)n,j (u).



Proposition. Bivariate joint probabilities for (τ, jUτj).For n = 1, we have

ξ(k1)1,j (u) = f (u + c + j ; k1).

where

f (j ; k1) =

8<: Pr�B1 + ...+ Bk1 = j

�, if k1 > 0

1, if k1 = 0 and j = 00, if k1 = 0 and j 6= 0

.


ξ(k1)n,j (u) =

∞

∑k2=0

Pr (N2 = k2 jN1 = k1)u+c

∑l=0

f (l , k1)ξ(k2)n�1,j (u + c � l).



Relations.

Ruin probabilities :

ψ(u, n) =n

∑k=1

∞

∑j=1

ξk ,j (u) .

Moments :

E [jUτ jm j τ � n,U(0) = u] =∑nk=1 ∑∞

j=1 jm ξk ,j (u)

ψ(u, n). (8)

More generally, for w (x , y) = ϕ (y) :

mδ(u, n) =n

∑k=1

∞

∑j=1

vk ϕ (j) ξk ,j (u) .


Trivariate joint probabilities for time at ruin, surplus beforeruin and de�cit at ruin

We de�ne

$(k1)n,i ,j (u) = Pr (τ = n,Uτ� = i , jUτj = j jU(0) = u,N1 = k1) ,

for n 2 N+, i = 0, 1, ..., j = 1, 2, ...

We have the following relationship

$n,i ,j (u) =∞

∑k1=0

pk1 � $(k1)n,i ,j (u).



Proposition. Trivariate joint probabilities for (τ,Uτ�, jUτj).For n = 1, we have

$(k1)1,i ,j (u) = δi ,u � f (u + π + j , k1),

where δi ,u = Kronecker delta of (i , u)

δi ,u =

�1, if i = u0, otherwise

.


$(k1)n,i ,j (u) =

∞

∑k2=0

pk1,k2u+c

∑l=0

f (l , k1)$(k2)n�1,i ,j (u + π � l).

where

f (j ; k1) =

8<: Pr�B1 + ...+ Bk1 = j

�, if k1 > 0

1, if k1 = 0 and j = 00, if k1 = 0 and j 6= 0

.



Advantage :

We have all the probabilities to compute any quantities of interestGeneral relation :

mδ(u, n) =n

∑k=1

∞

∑i=0

∞

∑j=1

vkw(i , j) $k ,i ,j (u) , for u � 0 and n 2 N+


In�nite-time ruin measures

Expected discounted penalty function :

mδ(u) = E�v τw (Uτ� , jU(τ)j) 1fτ<∞gjU(0) = u

�For computation purposes, we assume that N is a discrete-timeMarkov Chain de�ned on Am = f0, 1, 2, ...,mg.We compute mδ(u) with

mδ(u) =m

∑k1=0

pk1 �mδ(u;N1 = k1), for u 2 N, (9)

where

mδ(u;N1 = k1) = E�v τw (Uτ� , jU(τ)j) 1fτ<∞gjU(0) = u,N1 = k1

�,

(10)for k1 2 Am



Recursive relation for mδ(u;N1 = k1) (u 2 N) :

mδ(u;N1 = k1) = vm

∑k2=0

pk1,k2u+c

∑j=0

f (j ; k1)mδ(u + π � j ;N1 = k2)

+ vγ(u + π;N1 = k1), for k1 2 Am ,

where

γ(u;N1 = k1) =∞

∑j=u+1

w(u � π, j � u)f (j ; k1),

and

f (j ; k1) =


.



Two approaches to compute the values mδ(u;N1 = k1)

Approach #1 : Laplace TransformApproach #2 : Recursive relation

In both cases, we need to �nd the values of mδ(u;N1 = k1) foru 2 f0, 1, ..., c � 1g and k1 2 AmIn the case when N is a Poisson INAR(1) process, we approximate Nby a Markov chain N (m) de�ned on Am .

The m�truncated joint probability are given by

Pr(N (m)1 = k1,N(m)2 = k2) =

Pr(N (m)1 = k1,N(m)2 = k2)

Pr(N (m)1 � m,N (m)2 � m), k1, k2 2 Am .

m is �xed such that Pr(N1 � m,N2 � m) is close to 1


Conclusion

Examine the impact of dependence of various (�nite- andin�nite-time) ruin measures.

When the claim amount Bc is a continuous rv,

approximate Bc by a discrete rv B using discretization technicsuse the recursive formulas

Computation time can be signi�cative.


Epilogue �Numerical evaluation of ruin probabilities basedon simulation

Here is a brief comment about another important Gerber�scontribution : application of theory of martingale in ruin theoryWe consider continuous-time risk models with dependence (see e.g.e.g. Albrecher et Teugels (2006), Boudreault et al. (2007), Cossetteet al. (2009))We consider an insurance portfolioNotations

continuous rv Xk : amount of the kth claim (k 2 N+)continuous rv Wk : time elapsed between claim #(k � 1) and claim#k (k 2 N+)

Illustration0j0��| {z }W1

X1jT1��| {z }

W2

X2jT2��|{z}W3

X3jT3��| {z }

W4

X4jT4�|{z}W5

X5jT5�



Assumption : f(Xk ,Wk ) , k 2 N+g forms a sequence of i.i.d. pairs ofrvs with (Xk ,Wk ) � (X ,W ) for k 2 N+.

Joint pdf of (X ,W ) : fX ,W (x ,w)

Joint cdf of (X ,W ) : FX ,W (x ,w)

Joint mgf of (X ,W ): MX ,W (x ,w)

Surplus process : U = fU (t) , t � 0g où

U (t) = u + ct �∑N (t)k=1 Xk , t > 0

U (0) = u = initial surplus

c = premium rate where c = (1+ η)E [X ]E [W ]

with η > 0

Claim number process N = fN (t) , t � 0g whereN (t) = ∑∞

k=1 1fW1+...+Wk�tg



Random walk : Y = fYk , k 2 Ng where Y0 = 0 and

Yk = Yk�1 + (Xk � cWk ) , k = 1, 2, ...

In�nite-time ruin probability : ψ (u) = Pr (Z > u) whereZ = max

k2NfYkg.

Lundberg�s equation : Eher (X�cW )

i= MX ,W (r ,�cr)

Adjustment coe¢ cient : ρ is the unique strictly positive solution of

Eher (X�cW )

i= E

herLi= MX ,W (r ,�cr) = 1

where L = X � cW = net loss rv



Time of ruin : rv τ

Using theory of martingale, Gerber obtained the the following result

Gerber�s exact expression for ψ (u) :

ψ (u) =e�ρu

E�e�ρU (τ)jτ < ∞

� ,for u � 0.



3 approaches for numerical evaluation of ψ (u) based on simulation :

#1 : Crude Monte Carlo simulation method#2 : MC simulation method based on Gerber�s exact expression#3 : Importance sampling based change of measure

Nb of simulations : m

Approach #1 - Crude MC simulation : approximate ψ (u) by

eψ (u; ρ, n,m) = 1m

m

∑j=11fτ(j)<ng



Approach #2 - MC simulation based on Gerber�s exact expressionψ (u) = e�ρu

E [e�ρU (τ)jτ<∞]:

Approximate Ehe�ρU (τ)jτ < ∞

iby

∑mj=1 e�ρU (j)(τ(j))1fτ(j)<ng∑mj=1 1fτ(j)<ng

We obtain

eψ (u; ρ, n,m) = e�ρu1m ∑mj=1 1fτ(j)<ng

1m ∑mj=1 e

�ρU (j)(τ(j))1fτ(j)<ng

It seems to correct the bias in the approximation under approach #1



Approach #3 - Importance sampling

Based on another exact expression

ψ (u) = e�ρuE (ρ)heρU (τu )

ifor u � 0E (ρ) [...] : expectation de�ned on a new equivalent probability measure(see e.g. Asmussen & Albrecher (2009)).De�ne the rv L = X � cW with pdf fLThen, under new measure probability measure, we have

f (ρ)L (x) =eρx

E�eρL� fL (x)

for x 2 R.Di¢ culty : simulate under new probability measure



Example : Distribution of (X ,W ) is based on fgm copula

Cα (u1, u2) = u1u2 + θu1u2 (1� u1) (1� u2) ,

and X � Exp (β), W � Exp (λ),Joint pdf of (X ,W ) :

fX ,W (x1, x2) = βe�βx1λe�λx2

+θβe�βx1λe�λx2 + θ2βe�2βx12λe�2λx2

�θ2βe�2βx1λe�λx2 � θβe�βx12λe�2λx2 .



Example : (suite)

Joint pdf of (X ,W ) under change of measure : f (ρ)X ,W (x , y) = ...

(1+ θ)

E�eρL� � β

β� ρ

��λ

λ+ ρ

�(β� ρ) e�(β�ρ)x (λ+ ρ) e�(λ+ρ)y

� θ

E�eρL� � 2β

2β� ρ

��λ

λ+ ρ

�(2β� ρ) e�(2β�ρ)x (λ+ ρ) e�(λ+ρ)y

� θ

E�eρL� � β

β� ρ

��2λ

2λ+ ρ

�(β� ρ) e�(β�ρ)x (2λ+ ρ) e�(2λ+ρ)y

+θ

E�eρL� � 2β

2β� ρ

��2λ

2λ+ ρ

�(2β� ρ) e�(2β�ρ)x (2λ+ ρ) e�(2λ+ρ)y



Example : (suite)

β = 1, λ = 1, premium rate c = 1.5Dependence parameter θ = 0.5ρ = 0.3788264025

u ψ (u) exact ψ (u) Gerber ψ (u) IS0 0.64512257776 0.644885890756 0.644888070605 0.09495341322 0.095648195486 0.0947083302910 0.01428545356 0.014594674121 0.0145949997620 0.00032334926 0.000380799477 0.00032934350


Conclusion #2

Approach #2 (with Gerber�s formula) is easy to apply.

It is possible to apply both approaches #2 and #3 for many jointdistributions for (X ,W )

Bivariate exponential distributions (Downton-Moran, Raftery, etc.)Bivariate gamma distributionsBivariate mixed exponentialBivariate mixed ErlangEtc.

It is possible to examine various dependence structure and the impactof dependence on ψ (u)

Using results from Schmidli (2010), we use approach #3 to evaluateG-S function.

Work in progress ..

Thank you for your attention !


ruin theory in risk models based on time series for count random … · 2017-09-28 · ruin theory...

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