a fourier-cosine method for solving backward sdesthe markovian property, will be introduced. also, a...
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MSc Stochastics and Financial Mathematics
Master Thesis
A Fourier-Cosine method for solvingBackward SDEs
Author: Supervisor:
Caroline Kosalka dr. Asma Khedher (UvA)
Isabelle Liesker (RQ)
Examination date:
June 4, 2018
Korteweg-de Vries Institute forMathematics
RiskQuest B.V.
Abstract
Backward Stochastic Di↵erential Equations (BSDEs) are the stochastic counterpartof semi-linear parabolic di↵erential equations. Since the discovery of the generalizedFeynman-Kac formula, BSDEs gained a wide range of applications in economics andmore generally in optimal control. As it is not possible to solve BSDEs analytically, it islogical to ask for numerical methods approximating the unique solution of this type ofequations. This thesis presents the basic theory of BSDEs where the generator satisfiesthe Lipschitz condition. Moreover, a system of a BSDE associated with a state processsatisfying some forward SDE results in a so-called Forward-Backward SDE, satisfyingthe Markovian property, will be introduced. Also, a Fourier-Cosine method for com-puting the solutions of BSDEs, which proved to be very e�cient, is performed where acomputational example is discussed.
Title: A Fourier-Cosine method for solving Backward SDEsAuthor: Caroline Kosalka, [email protected], 10901396Supervisor: dr. Asma Khedher,Second Examiner: dr. Peter SpreijExamination date: June 4, 2018
Korteweg-de Vries Institute for MathematicsUniversity of AmsterdamScience Park 105-107, 1098 XG Amsterdamhttp://kdvi.uva.nl
RiskQuest B.V.Herengracht 495, 1017 BT Amsterdamhttp://www.riskquest.com
2
Acknowledgements
This thesis marks one of the final steps in obtaining the Master of Science degree inStochastics and Financial Mathematics at the University of Amsterdam (UvA). Theresearch was conducted during an internship at RiskQuest.First of all, I would like to thank RiskQuest for this opportunity as well as all my
colleagues for their support. I would like to thank my supervisor, Asma Kheder, for herguidance, especially during the last months of my thesis. Without her support I wouldnot have been able to go through this very challenging master thesis. Measure theoryand Stochastic Integration are one of the most abstract and theoretical courses in mymasters and therefore the thesis was a big challenge for me.I would like to thank also my friend Jenya for the late working hours in the library,
as well as for motivating me to go swimming. Sometimes presence is all what you need.Finally, my last thanks goes to my godmother, Justine, who supported me during mystudies and who made it possible for me to study abroad for so many years. I am gratefulfor all the di↵erent places, experiences and people I met, it definitely made my life morediverse as well as complete.
Dla mojego ojca, ktory nigdy nie przesta l byc tata.
3
Contents
Introduction 6
1. Backward Stochastic Di↵erential Equations 81.1. Motivation and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 81.2. Existence and Uniqueness of a solution to Backward Stochastic Di↵eren-
tial Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3. Linear BSDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4. Comparison Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.5. Applications of BSDEs in finance . . . . . . . . . . . . . . . . . . . . . . . 23
1.5.1. Stochastic Optimal Control Problem . . . . . . . . . . . . . . . . . 231.5.2. Hedging in the Black-Scholes model . . . . . . . . . . . . . . . . . 25
2. FBSDEs and Feynman-Kac formula 282.1. Forward-Backward Stochastic Di↵erential Equations . . . . . . . . . . . . 28
2.1.1. Regularity Properties of Solutions . . . . . . . . . . . . . . . . . . 292.1.2. Markov Properties of Solutions . . . . . . . . . . . . . . . . . . . . 30
2.2. The link with PDEs: Feynman-Kac formula . . . . . . . . . . . . . . . . . 30
3. Numerical Methods for BSDEs 343.1. Discretization of the SDE . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.1. Euler-Maruyama scheme . . . . . . . . . . . . . . . . . . . . . . . . 353.1.2. The Milstein Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.3. The weak Taylor Scheme of order 2.0 . . . . . . . . . . . . . . . . . 36
3.2. Discretization of the BSDE . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4. The Fourier-Cosine method for Backward Stochastic Di↵erential Equa-tions 424.1. Fourier Transform & Fourier Cosine series . . . . . . . . . . . . . . . . . . 424.2. Approximations of the conditional expectations . . . . . . . . . . . . . . . 46
4.2.1. COS approximation of the function Z
⇧i
. . . . . . . . . . . . . . . 494.2.2. COS approximation of the function Y
⇧i
. . . . . . . . . . . . . . . 494.2.3. Fourier-cosine coe�cients . . . . . . . . . . . . . . . . . . . . . . . 504.2.4. Discretization of the Fourier-cosine coe�cient . . . . . . . . . . . . 50
4.3. BCOS summarized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5. Numerical experiment: Hedging in the Black-Scholes model 52
A. Appendix 54
4
Conclusion 57
Popular summary 58
5
Introduction
Di↵erential equations first came into existence in 1671, where Isaac Newton solved dif-ferential equations using infinite series and discussed the non-uniqueness of solutions.They are used in describing di↵erent phenomena mathematically, for instance, some canbe modelled by ordinary di↵erential equations (ODE)
dx(t)
dt= f(t, x),
which is applied in di↵erent branches of science. However, noise, which is a sort ofrandomness can not be omitted. So a better approximation to the reality is
dx(t)
dt= f(t, x) + ’white noise’, (0.1)
where ’white’ means that the noise contains frequencies analogously to white light. Thusthis leads us to the mathematical concept of stochastic di↵erential equations (SDE).
In 1827, Robert Brown discovered the random movement of particles suspended ina fluid: Brownian motion. The mathematics behind Brownian motion were studied bymany great mathematicians, among others Norbert Wiener after whom Wiener processis named. This stochastic process is denoted W
t
where t � 0. The sample paths of aBrownian motion are continuous but proved to be nowhere di↵erentiable and even ofunbounded variation, so classic calculus fails to define integration with respect to it.Nevertheless for describing white noise it is the best candidate.
In 1940, Kiyosi Ito defined Ito stochastic integral and proved the Ito isometry (LemmaA.7).So the above equation (0.1) can be expressed as the following SDE
dx(t,!) = f(t, x)dt+ dWt
.
The notion of time is of big importance for SDEs, in contradiction to ODEs, where initialvalue problem and terminal value problem are treated in the same way.Solving an SDE with known terminal value implies the manipulation of a system to
achieve a prescribed aim in a randomly disturbed circumstance.A typical SDE on the time interval [0, T ] has the form
(dX
t
= b(t,Xt
)dt+ �(t,Xt
)dWt
,
X0 = x.
6
This SDE can be solved by Picard iteration and Banach fixed point theorem and thereexists a unique adapted solution under certain conditions.
However, this routine fails to give adapted solutions when the SDE is defined via theterminal value X
T
= ⇠ 2 FT
, known as the Backward stochastic di↵erential equation(BSDE).In 1973, the BSDEs were first introduced by J.M. Bismut [2] in the case where f is
linear w.r.t. (Y, Z). He used BSDEs to study stochastic optimal control problems in thestochastic version of the Pontryagin’s maximum principle.In 1990, the theory of BSDEs was a main focus for many academic researchers and
a large number of publications have been made. However the most famous authors arePardoux and Peng [19], who studied the BSDEs of the following form,
(�dY
t
= f(t, Yt
, Z
t
)dt� Z
t
dWt
,
Y
T
= ⇠,
(0.2)
and proved the existence and uniqueness of a solution to the latter BSDE where the gen-erator satisfies the Lipschitz condition. In 1992, Peng [23] introduced the comparisontheorem which provides a su�cient condition for the wealth process to be nonnegative.In [20], Pardoux and Peng showed that the solution of the BSDE in the Markovian casecorresponds to a probabilistic solution of a non-linear PDE, and gave a generalization ofthe Feynman Kac formula. Since the discovery of the generalized Feynman-Kac formula,applications in mathematical finance and economics started to gain a lot of attention.Moreover in [22], [25], [24], Peng stated the connection between the BSDEs associatedwith a state process satisfying some forward classical SDEs, and PDEs in the Markoviancases. The theories behind can be used for European option pricing in the constrainedMarkovian case.In 1995, in [9] the theory of contingent claim valuation, particularly the Black-Scholes-Merton model for option pricing is expressed in terms of a solution to a BSDE.In 2000, Kohlmann and Zhou [15], developed Peng’s stochastic optimal control theoryand interpreted BSDEs as equivalent to stochastic control problems.
Nowadays, Backward Stochastic Di↵erential Equations (BSDEs) remain an interestingfield attracting lots of well-known researchers’ investigation. Therefore, it is interestingto know how the theory of BSDE and its applications is being developed.This thesis is organized as follows. In Chapter 1 we provide a general introduction to BS-DEs and discuss uniqueness and existence results as well as the Comparison Theorem.In Chapter 2 Forward-Backward SDEs with their Markovian property are introducedand the generalized Feynman-Kac theorem is proven in both directions. This gives usan important link between BSDEs and PDEs. To obtain a numerical solution to a FB-SDE we present in Chapter 3 discretization methods for SDEs and introduce the theta-discretization for BSDEs which results in expressions with conditional expectations. Tocompute these conditional expectations, we develop, in Chapter 4, a Fourier-Cosinemethod, called BCOS method. Finally, in Chapter 5 a computational example is given.
7
1. Backward Stochastic Di↵erentialEquations
The aim of this chapter is to get familiar with the notion of Backward Stochastic Dif-ferential Equations (BSDEs). After introducing the notation and the main definitionsof BSDEs, we will prove two important theorems: the existence and uniqueness of asolution for a BSDE satisfying the Lipschitz condition and the Comparison Theorem.We will finish this chapter by providing some examples. The following references [9],[18] are used.
1.1. Motivation and preliminaries
Let (⌦,F ,P) be a complete probability space, where the R-valued standard Brownianmotion W is defined with respect to the natural filtration F
t
. We consider a simpleordinary di↵erential equation (ODE)
dYt
= 0, 0 t T, (1.1)
where T > 0 is a given terminal time. For any ⇠ 2 R we can suppose either Y0 = ⇠
or Y
T
= ⇠, such that the above ODE has a unique solution Y
t
= ⇠. However, if theequation (1.1) is considered as a Stochastic Di↵erential Equation (SDE), we are in adi↵erent setting. The solution of (1.1) seen in a stochastic sense should be adapted tothe filtration {F
t
}t�0. Therefore specifying Y
T
or Y0 does a big di↵erence.Consider the ODE (1.1) with the following terminal condition:
(dY
t
= 0 0 t T,
Y
T
= ⇠,
(1.2)
where ⇠ is a random FT
-measurable and square-integrable variable. Since (1.2) is anODE with a unique solution given by Y
t
= ⇠, which is not necessarily adapted to thefiltration F
t
, unless ⇠ is constant, the equation (1.2) viewed as an SDE, does not have asolution in general. Note that in the case where an initial condition is given, the solutionis adapted.To deal with this terminal value problem we will reformulate (1.2) in such a way that wecan ensure the adaptability of the solution to F
t
. We introduce the following conditionalexpectation,
Y
t
:= E (⇠|Ft
) 0 t T. (1.3)
8
Since ⇠ is FT
-adapted we have YT
= ⇠. As the process (Yt
)t�0 defined by (1.3) is square-
integrable Ft
�martingale, we get by the Martingale Representation Theorem (A.2),
Y
t
= Y0 +
Zt
0Z
s
dWs
, 0 t T a.s., (1.4)
where Z
s
is a Ft
�adapted square integrable process. From (1.3) and (1.4) we get,(dY
t
= Z
t
dWt
0 t T
Y
T
= ⇠.
(1.5)
In other words, we reformulated (1.2) by (1.5) and what is of big importance is thatinstead of seeking only for one F
t
-adapted stochastic process Y as a solution to theSDE, we are looking for a pair (Y, Z). This method by adding the extra component Zto the solution makes it possible to find an F
t
-adapted solution.We will rewrite the backward SDE (1.5) in an integral form. To do this, the SDE is firstevaluated in (1.4) with the terminal condition Y
T
= ⇠ and solved for Y0.
Y0 = Y
T
�Z
T
0Z
s
dWs
= ⇠ �Z
T
0Z
s
dWs
. (1.6)
Plugging (1.6) into (1.4) we get
Y
t
= ⇠ �Z
T
t
Z
s
dWs
0 t T. (1.7)
Hence we have established the idea of how Backward SDEs appeared and that they canbe represented by (1.5) or (1.7).We introduce the following notations which will be used throughout the thesis:
• T > 0, the so-called terminal time.
• {Ft
; t 2 [0, T ]}, the filtration generated by the Brownian motion W and augmentedby all the P-null sets.
• P the �-field of predictable sets on ⌦⇥ [0, T ].
• B = B(R), the Borel �-algebra and the Lebesgue measure on B is denoted by �.
• L2T
(R), the space of all FT
-measurable random variables X : ⌦ ! R satisfyingkXk2 = E |X|2 < 1.
• H↵
T
(R), the space of all predictable processes : ⌦ ⇥ [0, T ] ! R such that
E✓qR
T
0 | t
|2 dt◆
↵
< 1, where ↵ > 0.
• For fixed � 2 R+ and 2 H2T
(R), || ||2�
:= E⇣R
T
0 e
�t| t
|2dt⌘. H2
T,�
(R) denotesthe space H2
T
(R) endowed with the norm || · ||�
.
9
For notational simplicity we sometimes use: L2T
(R) = L2T
, H↵
T
(R) = H↵
T
, H2T,�
(R) = H2T,�
and for f(!, t, Yt
, Z
t
) = f(t, Yt
, Z
t
).
Definition 1.1. Let ⇠ : ⌦! R be an FT
-measurable random variable and letf : ⌦⇥ [0, T ]⇥ R⇥ R ! R be a P ⌦ B ⌦ B-measurable mapping. Then,
(�dY
t
= f(!, t, Yt
, Z
t
)dt� Z
t
dWt
; t 2 [0, T ],
Y
T
= ⇠,
(1.8)
or, equivalently,
Y
t
= ⇠ +
ZT
t
f(!, s, Ys
, Z
s
)ds�Z
T
t
Z
s
dWs
t 2 [0, T ] (1.9)
is called a Backward Stochastic Di↵erential Equation (BSDE) with terminal value ⇠ andgenerator f .
Properties of di↵erent BSDEs, such as, linear BSDEs, Lipschitz BSDEs, MarkovianBSDEs, quadratic BSDEs, etc. are properties of the generator f . In other words a linearBSDE represents a BSDE with a linear generator, i.e. f(t, Y
t
, Z
t
) = ↵
t
+ �
t
Y
t
+ �
t
Z
t
. Inthis thesis we will only consider Lipschitz BSDEs, unless stated otherwise.
Definition 1.2. Suppose that ⇠ 2 L2T
(R), f(·, 0, 0) 2 H2T
(R) and assume f is uniformlyLipschitz continuous; i.e. there exists constant C > 0 such that dP⌦ dt-a.s.
|f(t, y1, z1)� f(t, y2, z2)| C (|y1 � y2|+ |z1 � z2|) ,
holds true 8(y1, z1), (y2, z2) 2 R⇥R. Then the pair (f, ⇠) is said to be a set of standardparameters of the BSDE (1.8).
Definition 1.3. A solution to the BSDE (1.8) is a pair of value (Y, Z) such that {Yt
; t 2[0, T ]} is a continuous R-valued (F
t
)t�0-adapted process and {Z
t
; t 2 [0, T ]} is an R-valued predictable process satisfying
RT
0 |Zs
|2 ds < 1, P-a.s..
Note that a solution (Y, Z) 2 H2T
⇥ H2T
, when it exists, is often referred to a square-integrable solution.
1.2. Existence and Uniqueness of a solution to BackwardStochastic Di↵erential Equation
In this section we present in detail the existence and uniqueness of a solution to theBSDE (1.8). Following the derivations in [9] we first introduce a priori estimates thatwe need to prove the main Theorem 1.5.
10
Lemma 1.4. [3] Let ((f i
, ⇠
i); i = 1, 2) be two standard parameters of the BSDE (1.8)and ((Y i
, Z
i); i = 1, 2) be two square-integrable solutions to the corresponding standardparameters. Consider C be the Lipschitz constant of f1 and denote
�Y
t
= Y
1t
� Y
2t
�Z
t
= Z
1t
� Z
2t
�2ft = f
1(t, Y 2t
, Z
2t
)� f
2(t, Y 2t
, Z
2t
).
Then for any (�, µ,�) such that µ > 0,�2 > C and � � C(2 + �
2) + µ
2, it follows that
k�Y k2�
T
e
�TE(|�YT
|2) + 1
µ
2k�2fk2
�
�, (1.10)
k�Z|2�
�
2
�
2 � C
e
�TE(|�YT
|2) + 1
µ
2k�2fk2
�
�. (1.11)
Proof. We divide the proof into four steps.
1. Let (Y, Z) 2 H2T
⇥ H2T
be a solution of the BSDE where (f, ⇠) are the standardparameters,
Y
t
= ⇠ +
ZT
t
f(s, Ys
, Z
s
)ds�Z
T
t
Z
s
dWs
t 2 [0, T ]. (1.12)
In this step we want to show that sup0tT
|Yt
| 2 L2T
. To show this we will takethe absolute value on both sides as well as the supremum and show that the righthand side of equation (1.12) is in L2
T
. Taking the absolute value on both sides ofequation (1.12), we have
|Yt
| =����⇠ +
ZT
t
f(s, Ys
, Z
s
)ds�Z
T
t
Z
s
dWs
����
|Yt
| |⇠|+����Z
T
t
f(s, Ys
, Z
s
)ds
����+����Z
T
t
Z
s
dWs
���� (by the triangle inequality)
|Yt
| |⇠|+Z
T
t
|f(s, Ys
, Z
s
)| ds+����Z
T
t
Z
s
dWs
���� (by Jensen inequality)
sup0tT
|Yt
| |⇠|+Z
T
0|f(s, Y
s
, Z
s
)| ds+ sup0tT
����Z
T
t
Z
s
dWs
���� . (1.13)
By assumption, ⇠ 2 L2T
, i.e. E(|⇠|2) < 1 so we have that |⇠| 2 L2T
. For the secondterm in (1.13) we will use the Lipschitz continuity of f , we obtain,
|f(s, Ys
, Z
s
)| |f(s, Ys
, Z
s
)� f(s, 0, 0)|+ |f(s, 0, 0)| C(|Y
s
|+ |Zs
|) + |f(s, 0, 0)|. (?)
11
Moreover, we have,
E✓Z
T
0|f(s, Y
s
, Z
s
)| ds◆2
TE✓Z
T
0|f(s, Y
s
, Z
s
)|2 ds◆
(by Cauchy-Schwarz inequality)
TE✓Z
T
0(C(|Y
s
|+ |Zs
|) + |f(s, 0, 0)|)2 ds◆
(by (?))
4TC2E✓Z
T
0|Y
s
|2ds◆+ 4TC2E
✓ZT
0|Z
s
|2ds◆
+ 2TE✓Z
T
0|f(s, Y
s
, Z
s
)|2 ds◆
( by Lemma A.5 )
< 1.
HenceRT
0 |f(s, Ys
, Z
s
)| ds 2 L2T
. For the third term in (1.13),we get
E
sup
0tT
����Z
T
t
Z
s
dWs
����2!
= E
sup
0tT
����Z
T
0Z
s
dWs
�Z
t
0Z
s
dWs
����2!
2E
����Z
T
0Z
s
dWs
����2!
+ 2E
sup
0tT
����Z
t
0Z
s
dWs
����2!
(by Lemma A.5)
2E
����Z
t
0Z
s
dWs
����2!
+ 2E
sup
0tT
����Z
t
0Z
s
dWs
����2!.
By Ito’s isometry (A.7) we have:
E
����Z
t
0Z
s
dWs
����2!
= E✓Z
t
0|Z
s
|2ds◆.
By Burkholder-Davis-Gundy inequality (A.1), there exists a constant K > 0 suchthat
E
sup
0tT
����Z
t
0Z
s
dWs
����2!
KE✓Z
t
0|Z
s
|2ds◆.
So for the third term in (1.13), we get
E
sup
0tT
����Z
T
t
Z
s
dWs
����2!
2(1 +K)E✓Z
t
0|Z
s
|2ds◆
< 1.
Thus we showed sup0tT
���RT
t
Z
s
dWs
��� 2 L2T
, which leads to sup0tT
|Yt
| 2 L2T
.
12
2. Consider the two solutions (Y 1, Z
1) and (Y 2, Z
2) of the BSDE (1.8) with thestandard parameters (f1
, ⇠
1) and (f2, ⇠
2), respectively. By Ito formula applied toe
�s |�Ys
|2, one obtains
d⇣e
�s |�Ys
|2⌘= �e
�s |�Ys
|2 ds+ 2e�s�Ys
d�Ys
+ e
�sd�Ys
d�Ys
.
Integrating the above equation from t to T , we have
e
�t |�Yt
|2 + �
ZT
t
e
�s |�Ys
|2 ds+Z
T
t
e
�s |�Zs
|2 ds
=e
�T |�YT
|2 + 2
ZT
t
e
�s
�Y
s
�f
1(s, Y 1s
, Z
1s
)� f
2(s, Y 2s
, Z
2s
)�ds (1.14)
� 2
ZT
t
e
�s
�Y
s
�Z
s
dWs
.
3. In this step, we want to show
E⇣e
�t |�Yt
|2⌘ E
⇣e
�T |�YT
|2⌘+
1
µ
2E✓Z
T
t
e
�s |�2fs|2 ds◆
t 2 [0, T ].
From step 1, we know sup0tT
|Yt
| 2 L2T
, which leads to
E
0
@sZ
T
0|e�s�Z
s
�Y
s
|2ds1
A
= E
0
@sZ
T
0e
2�s|�Zs
|2|�Ys
|2ds1
A
E
0
@ sup0tT
|�Yt
|sZ
T
0e
2�s|�Zs
|2ds1
A
vuutE
sup
0tT
|�Yt
|2!s
E✓Z
T
0e
2�s|�Zs
|2ds◆
(by Cauchy Schwarz inequality)
< 1.
We showed that e�s�Zs
�Y
s
belongs to H1T
which implies that the stochastic integralRT
t
e
�s
�Y
s
�Z
s
dWs
becomes P-integrable and has zero expectation. Moreover,��f
1(s, Y 1s
, Z
1s
)� f
2(s, Y 2s
, Z
2s
)��
=��f
1(s, Y 1s
, Z
1s
)� f
1(s, Y 2s
, Z
2s
) + f
1(s, Y 2s
, Z
2s
)� f
2(s, Y 2s
, Z
2s
)��
��f1(s, Y 1s
, Z
1s
)� f
1(s, Y 2s
, Z
2s
)��+��f
1(s, Y 2s
, Z
2s
)� f
2(s, Y 2s
, Z
2s
)��
C (|�Ys
|+ |�Zs
|) + |�2fs|
13
and
2��h�Y
s
, f
1(s, Y 1s
, Z
1s
)� f
2(s, Y 2s
, Z
2s
)i�� 2 |�Y
s
| ��f1(s, Y 1s
, Z
1s
)� f
2(s, Y 2s
, Z
2s
)��
2C |�Ys
|2 + 2 |�Ys
| (C |�Zs
|+ |�2fs|),where C is the Lipschitz constant. Taking the expectation in (1.14) we get:
E⇣e
�t |�Yt
|2⌘+ �E
✓ZT
t
e
�s |�Ys
|2 ds◆+ E
✓ZT
t
e
�s |�Zs
|2 ds◆
=E⇣e
�T |�YT
|2⌘+ 2E
✓ZT
t
e
�s
�Y
s
(f1(s, Y 1s
, Z
1s
)� f
2(s, Y 2s
, Z
2s
))ds
◆. (??)
We take the last term in the above equation and we rewrite as follows,
E✓Z
T
t
2e�s�Ys
(f1(s, Y 1s
, Z
1s
)� f
2(s, Y 2s
, Z
2s
))ds
◆
E✓Z
T
t
2e�s|�Ys
||f1(s, Y 1s
, Z
1s
)� f
2(s, Y 2s
, Z
2s
)|ds◆
E✓Z
T
t
2e�sC |�Ys
|2 + 2e�s |�Ys
| (C |�Zs
|+ |�2fs|)ds◆
E✓Z
T
t
2e�sC |�Ys
|2 + e
�s
✓C
|�Zs
|2�
2+
|�2fs|2µ
2+ |�Y
s
|2(µ2 + C�
2)
◆ds
◆
(by Lemma A.6)
=�C(2 + �
2) + µ
2�E✓Z
T
t
e
�s |�Ys
|2 ds◆+
C
�
2E✓Z
T
t
e
�s |�Zs
|2 ds◆
+1
µ
2E✓Z
T
t
e
�s |�2fs|2 ds◆.
Now equation (??) becomes
E⇣e
�t |�Yt
|2⌘+ �E
✓ZT
t
e
�s |�Ys
|2 ds◆+ E
✓ZT
t
e
�s |�Zs
|2 ds◆
E⇣e
�T |�YT
|2⌘+�C(2 + �
2) + µ
2�E✓Z
T
t
e
�s |�Ys
|2 ds◆
(1.15)
+C
�
2E✓Z
T
t
e
�s |�Zs
|2 ds◆+
1
µ
2E✓Z
T
t
e
�s |�2fs|2 ds◆.
By rearranging the terms in (1.15) we get
E⇣e
�t |�Yt
|2⌘E
⇣e
�T |�YT
|2⌘+�C(2 + �
2) + µ
2 � �
�E✓Z
T
t
e
�s |�Ys
|2 ds◆
+
✓C
�
2� 1
◆E✓Z
T
t
e
�s |�Zs
|2 ds◆+
1
µ
2E✓Z
T
t
e
�s |�2fs|2 ds◆.
14
The above inequality with � � C(2 + �
2) + µ
2 and C < �
2, becomes
E⇣e
�t |�Yt
|2⌘ E
⇣e
�T |�YT
|2⌘+
1
µ
2E✓Z
T
t
e
�s |�2fs|2 ds◆. (1.16)
4. In this step we integrate (1.16) on both sides from 0 to T and we use Fubini’stheorem to change the order of the integrals, we get
E✓Z
T
0e
�t |�Yt
|2 dt◆
TE⇣e
�T |�YT
|2⌘+
1
µ
2
ZT
0E✓Z
T
t
e
�s |�2fs|2 ds◆dt
T
✓e
�TE⇣|�Y
T
|2⌘+
1
µ
2E✓Z
T
t
e
�s |�2fs|2 ds◆◆
.
This is equivalent to
k�Y k2�
T
e
�TE |�YT
|2 + 1
µ
2k�2fk2
�
�.
By rearranging the terms in (1.15), we can also derive
✓�
2 � C
�
2
◆EZ
T
t
e
�s |�Zs
|2 ds E⇣e
�T |�YT
|2⌘� E
⇣e
�t |�Yt
|2⌘
+�C(2 + �
2) + µ
2 � �
�E✓Z
T
t
e
�s |�Ys
|2 ds◆
+1
µ
2E✓Z
T
t
e
�s |�2fs|2 ds◆.
Now, the above inequality with � � C(2 + �
2) + µ
2 and C < �
2, becomes
✓�
2 � C
�
2
◆EZ
T
t
e
�s |�Zs
|2 ds E⇣e
�T |�YT
|2⌘+
1
µ
2E✓Z
T
t
e
�s |�2fs|2 ds◆.
Notice that, we can take � > 1 and t = 0, which leads to
k�Zk2�
�
2
�
2 � C
e
�TE(|�YT
|2) + 1
µ
2k�2fk2
�
�.
Theorem 1.5. [3] Given standard parameters (f, ⇠) there exists a unique pair (Y, Z) 2H2
T
(R)⇥H2T
(R) which solves the BSDE (1.8).
Proof. We will use the Banach fixed-point theorem for the following mapping:
: H2T,�
⇥H2T,�
! H2T,�
⇥H2T,�
(U, V ) 7! (Y, Z)
15
where (Y, Z) is a solution of the BSDE (1.8) with generator f(s, Us
, V
s
):
Y
t
= ⇠ +
ZT
t
f(s, Us
, V
s
)ds�Z
T
t
Z
s
dWs
0 t T. (1.17)
We divide the proof into steps.
1. We will show that is indeed well defined, i.e. there exists a solution (Y, Z) to(1.17). The assumption (f, ⇠) being standard parameters implies that f(·, U, V ) 2H2
T
E✓Z
T
0|f(s, U
s
, V
s
)|2ds◆
4C2E✓Z
T
0|U
s
|2ds◆+ 4C2E
✓ZT
0|V
s
|2ds◆
+ 2E✓Z
T
0|f(s, 0, 0)|2ds
◆
<1.
For t 2 [0, T ] we have,
E
����Z
T
t
f(s, Us
, V
s
)ds
����2!
(T � t)E✓Z
T
t
|f(s, Us
, V
s
)|2ds◆
TE✓Z
T
0|f(s, U
s
, V
s
)|2ds◆
<1.
Hence we showed thatRT
t
f(s, Us
, V
s
)ds 2 L2T
8t 2 [0, T ]. Consider
M
t
:= E✓Z
T
0f(s, U
s
, V
s
)ds+ ⇠
����Ft
◆,
which is a square-integrable martingale. Indeed, notice that since 8s t, Fs
⇢ Ft
,we have,
E (Mt
|Fs
) = E✓E✓Z
T
0f(s, U
s
, V
s
)ds+ ⇠
��Ft
◆ ����Fs
◆(by the tower property)
= E✓Z
T
0f(s, U
s
, V
s
)ds+ ⇠
����Fs
◆
= M
s
.
By the Martingale Representation Theorem (A.2), we know there exists a uniqueintegrable process Z 2 H2
T
such that
M
t
= M0 +
Zt
0Z
s
dWs
, t 2 [0, T ]. (1.18)
16
Next we define the adapted and continuous process Y as
Y
t
:= M
t
�Z
t
0f(s, U
s
, V
s
)ds.
We obtain
Y
t
= E✓Z
T
0f(s, U
s
, V
s
)ds+ ⇠
����Ft
◆�Z
t
0f(s, U
s
, V
s
)ds
= E✓Z
T
t
f(s, Us
, V
s
)ds+ ⇠
����Ft
◆.
Now it is easy to show that Y 2 H2T
,
E✓Z
T
0|Y
t
|2dt◆
= E
ZT
0
����E✓Z
T
t
f(s, Us
, V
s
)ds+ ⇠
����Ft
◆����2
dt
!
=
ZT
0E
����E✓Z
T
t
f(s, Us
, V
s
)ds+ ⇠
����Ft
◆����2!dt
(by Fubini’s theorem)
Z
T
0E
E����Z
T
t
f(s, Us
, V
s
)ds+ ⇠
����2 ����Ft
!dt
Z
T
0E
����Z
T
t
f(s, Us
, V
s
)ds+ ⇠
����2!dt
2
ZT
0E
����Z
T
t
f(s, Us
, V
s
)ds
����2
+ |⇠|2!dt (by Lemma A.5)
< 1.
In order to conclude this step, we need to clarify that the unique determined pair(Y, Z) solves the equation (1.17). But
⇠ = Y
T
= M
T
�Z
T
0f(s, U
s
, V
s
)ds
= M0 +
ZT
0Z
s
dWs
�Z
T
0f(s, U
s
, V
s
)ds. (1.19)
Using (1.18) and (1.19), we deduce
Y
t
= M
t
�Z
t
0f(s, U
s
, V
s
)ds
= M0 +
Zt
0Z
s
dWs
�Z
t
0f(s, U
s
, V
s
)ds (by using 1.18)
17
=
✓⇠ �
ZT
0Z
s
dWs
+
ZT
0f(s, U
s
, V
s
)ds
◆+
Zt
0Z
s
dWs
�Z
t
0f(s, U
s
, V
s
)ds
(by using 1.19)
= ⇠ +
ZT
t
f(s, Us
, V
s
)ds�Z
T
t
Z
s
dWs
and hence we get the claim.
2. Let (U1, V
1), (U2, V
2) be two elements of H2T,�
⇥ H2T,�
, and consider their images
(U1, V
1) = (Y 1, Z
1) and (U2, V
2) = (Y 2, Z
2) . We use the following notation
�2fs = f(s, U1s
, V
1s
)� f(s, U2s
, V
2s
)
By Lemma (1.4) applied with C = 0 and � = µ
2, we have
k�Y k2�
T
�
EZ
T
0e
�s
��f(s, U1
s
, V
1s
)� f(s, U2s
, V
2s
)��2 ds
T
�
C
2EZ
T
0e
�s (|�Us
|+ |�Vs
|)2 ds
2T
�
C
2�k�Uk2
�
+ k�V k2�
�
and
k�Zk2�
1
�
EZ
T
0e
�s
��f(s, U1
s
, V
1s
)� f(s, U2s
, V
2s
)��2 ds
2
�
C
2�k�Uk2
�
+ k�V k2�
�.
We deduce that
k�Y k2�
+ k�Zk2�
2(1 + T )C2
�
�k�Uk2�
+ k�V k2�
�. (1.20)
Now, let � > 2(1+T )C2, we see that the mapping is a contraction from H2T,�
⇥H2
T,�
onto itself and that there exists a unique fixed point (Y , Z) 2 H2T,�
⇥ H2T,�
satisfying
Y
t
= ⇠ +
ZT
t
f(s, Ys
, Z
s
)ds�Z
T
t
Z
s
dWs
0 t T.
We can choose the continuous version Y defined by
Y
t
:= E✓Z
T
t
f(s, Ys
, Z
s
)ds+ ⇠
����Ft
◆= E
✓ZT
t
f(s, Ys
, Z
s
)ds+ ⇠
����Ft
◆,
i.e. (Y, Z) is the unique continuous solution of the BSDE.
18
1.3. Linear BSDE
Linear BSDEs have a very useful property as it has an explicitly solution. In financialmathematics the solution of a linear BSDE is related to the pricing and hedging problemof a contingent claim.
Proposition 1.6. [21] Let (�, µ) be a bounded (R,R)-valued predictable process, ' anelement in H2
T
and ⇠ 2 L2T
. We consider the following linear BSDE:
Y
s
= ⇠ +
ZT
s
('u
+ �
u
Y
u
+ µ
u
Z
u
)du�Z
T
s
Z
u
dWu
, 0 t T (1.21)
a) The equation (1.21) has a unique solution (Y, Z) 2 H2T
⇥H2T
, where Y is given bythe following explicit formula
Y
t
= E⇠�t
T
+
ZT
t
�ts
'
s
ds|Ft
�, P� a.s.
where (�ts
)s�t
is the adjoint process defined by the forward linear SDE
d�ts
= �ts
(�s
ds+ µ
s
dWs
), �tt
= 1. (1.22)
b) If ⇠,' are non negative, then the process (Yt
)tT
is non negative. If in additionY0 = 0, then for any s � t, Y
s
= 0, ⇠ = 0 and 's
= 0, dP⌦ ds� a.s.
Proof. The first step is to show that (f, ⇠) are the standard parameters. ⇠ 2 L2T
is givenand f is uniformly Lipschitz by the below inequality,
|f(t, y1, z1)� f(t, y2, z2)| = |'t
+ �
t
y1 + µ
t
z1 � '
t
� �
t
y2 � µ
t
z2|= |�
t
(y1 � y2) + µ
t
(z1 � z2)| C|(y1 � y2) + (z1 � z2)|,
where |�t
| < K1, |µt
| < K2 and C = max(K1,K2). Since � and µ are boundedprocesses and the pair (f, ⇠) are standard parameters as well as the linear generatorf(t, y, z) = '
t
+ �
t
y
t
+ µ
t
z
t
satisfies the Lipschitz condition, by Theorem (1.5) thereexists a unique square-integrable solution (Y, Z) of the BSDE (1.21).As � and µ are bounded processes we also have the existence and uniqueness result forthe SDE (1.22) (i.e. see section 6.2. of [28] ).
In the proof of Theorem (1.5) we have already shown that E⇥sup0sT
|Ys
|⇤ < 1,
19
in other words that is sup0sT
|Ys
| 2 L2T
. We will show that Ehsup0sT
���ts
��2i< 1,
E
sup
0sT
���ts
��2!
=E
sup
0sT
�����t
t
+
Zt
0�
s
�ts
ds+
Zt
0µ
s
�ts
dWs
����2!
3E
sup
0sT
���tt
��2!
+ 3E
sup
0sT
����Z
t
0�
s
�ts
ds
����2!
+ 3E
sup
0sT
����Z
t
0µ
s
�ts
dWs
����2!
3 + 3E✓Z
T
0
���
s
�ts
��2 ds◆+ 3E
sup
0sT
����Z
t
0µ
s
�ts
dWs
����2!
(by Jensen inequality)
3 + 3E✓Z
T
0|�
s
|2 ���ts
��2 ds◆+ 3C
p
E✓Z
T
0
��µ
s
�ts
��2 ds◆
(by BDG inequality (A.1))
3 + 3E✓Z
T
0|�
s
|2 ���ts
��2 ds◆+ 3C
p
E✓Z
T
0|µ
s
|2 ���ts
��2 ds◆
3 + 3C(1 + C
p
)E✓Z
T
0
���ts
��2 ds◆
<1. (as���t
s
�� is square-intagrable)
By applying Ito’s formula to �ts
Y
s
, we get
d(�ts
Y
s
) = �ts
dYs
+ Y
s
d�ts
+ d�ts
dYs
= �ts
(�'s
ds� Y
s
�
s
ds� µ
s
Z
s
ds+ Z
s
dWs
) + �ts
Y
s
(�s
ds+ µ
s
dWs
) + �ts
Z
s
µ
s
ds
= �'s
�ts
ds+ �ts
(Zs
+ Y
s
µ
s
)dWs
.
Which implies that the process��ts
Y
s
+Rs
0 'u
�tu
du�tT
is a local martingale. If thelatter is a martingale, we will get
�tt
Y
t
+
Zt
0'
s
�ts
ds = E✓�tT
⇠ +
ZT
0'
s
�ts
ds
����Ft
◆
�tt
Y
t
= E✓�tT
⇠ +
ZT
t
'
s
�ts
ds
����Ft
◆,
which implies
Y
t
= E✓�tT
⇠ +
ZT
t
'
s
�ts
ds
����Ft
◆,
20
and the claim a) of the proposition is proved. Thus we need to prove that �ts
Y
s
+Rs
0 'u
�tu
du is a martingale. To do that we prove E⇥sup0sT
|�ts
Y
s
+Rs
0 'u
�tu
du|⇤ < 1.Notice that
E
"sup
0sT
�����t
s
Y
s
+
Zs
0'
u
�su
du
����
# E
"sup
0sT
���ts
Y
s
��#+ E
"sup
0sT
����Z
s
0'
u
�tu
du
����
#. (1.23)
From (1.23) we will first show that the first and second terms are finite, we have
E
"sup
0sT
���ts
Y
s
��# E
"sup
0sT
���ts
�� sup0sT
|Ys
|#
E
"sup
0sT
���ts
��2# 1
2
E
"sup
0sT
|Ys
|2# 1
2
(by Holder’s inequality)
< 1.
E
"sup
0sT
����Z
s
0'
u
�tu
du
����
# E
ZT
0
��'
u
�tu
�� du�
EZ
T
0|'
u
|2du� 1
2
EZ
T
0
���tu
��2 du� 1
2
(by Holder’s inequality)
T
12EZ
T
0|'
u
|2du� 1
2
E
"sup
0sT
���ts
��2# 1
2
< 1.
Hence (�ts
Y
s
+Rs
0 'u
�tu
du)tT
is a uniform-integrable local martingale which implies itis a martingale.b) Consider Y0 = 0 and use a), we get
�00Y0 +
Z 0
0'
s
�0s
ds = E✓�0T
⇠ +
ZT
0'
s
�0s
ds
����F0
◆
0 = E✓�0T
⇠ +
ZT
0'
s
�0s
ds
◆,
then the nonnegative variable �0T
⇠ +RT
0 '
s
�0s
ds has zero expectation. Therefore ⇠ = 0,P� a.s., '
s
= 0, dP⌦ dt� a.s. and Y = 0 a.s.. Particularly, if ⇠ and ' are nonnegative,Y
s
is nonnegative.
21
1.4. Comparison Theorem
This subsection is devoted to the comparison theorem which allows to compare thesolutions of two BSDEs as soon as one can compare the terminal conditions and thegenerators.
Theorem 1.7. [9] Let (Y, Z) and (Y , Z) be the solutions of two BSDEs with associatedparameters (f, ⇠) and (f , ⇠). We suppose that
⇠ ⇠, P-a.s.
andf(t, Y
t
, Z
t
)� f(t, Yt
, Z
t
) 0, dt⌦ dP -a.s.
Then we have that P�almost surely for any time t 2 [0, T ],
Y
t
Y
t
.
Moreover the comparison is strict, i.e., if in addition Y0 = Y0, then ⇠ = ⇠, f(t, Yt
, Z
t
) =f(t, Y
t
, Z
t
) and Y
t
= Y
t
, 8t 2 [0, T ] P� a.s.
In particular, whenever P(⇠ < ⇠) > 0 or f(t, Yt
, Z
t
)� f(t, Yt
, Z
t
) < 0 on a set of positivedt⌦ dP-measure, then Y0 < Y0.
Proof. We define: 8>>>><
>>>>:
'
t
= f(t, Yt
, Z
t
)� f(t, Yt
, Z
t
)
Y
0 = Y � Y
Z
0 = Z � Z
⇠
0 = ⇠ � ⇠.
For all t � 0,
Y
0t
= ⇠
0 +
ZT
t
(f(t, Yt
, Z
t
)� f(t, Yt
, Z
t
))ds�Z
T
t
Z
0s
dWs
We rewrite f as follows
f(t, Yt
, Z
t
)� f(t, Yt
, Z
t
) =f(t, Yt
, Z
t
)� f(t, Yt
, Z
t
) + f(t, Yt
, Z
t
)� f(t, Yt
, Z
t
)
+ f(t, Yt
, Z
t
)� f(t, Yt
, Z
t
).
Now we introduce two processes � and µ as follows
�
t
=
((Y
t
� Y
t
)�1(f(t, Yt
, Z
t
)� f(t, Yt
, Z
t
)), if Y
t
6= Y
t
,
0, otherwise,
µ
t
=
((Z
t
� Z
t
)�1(f(t, Yt
, Z
t
)� f(t, Yt
, Z
t
)), if Z
t
6= Z
t
,
0 otherwise.
22
By writing Y
0t
in terms of � and µ, we get
Y
0t
= ⇠
0 +
ZT
t
(Y 0s
�
s
+ Z
0s
µ
s
+ '
s
)ds�Z
T
t
Z
0s
dWs
.
The couple (Y 0t
, Z
0t
) is a solution of a linear BSDE. Note that (�, µ) satisfy the assump-tions of Proposition 1.6. Indeed �, µ are bounded as f is Lipschitz and the integrabilitycondition of the process ' can be deduced easily from the standards hypotheses andfrom the properties of Y and Z. From Proposition 1.6 we get 8t 2 [0, T ],
Y
t
= E⇠�t
T
+
ZT
t
�ts
'
s
ds|Ft
�,
where (�ts
)s�t
is the adjoint process defined by the following forward linear SDE
d�ts
= �ts
(�s
ds+ µ
s
dWs
).
Hence ⇠0 0 and 's
0, then Y
0t
0 follows.
1.5. Applications of BSDEs in finance
1.5.1. Stochastic Optimal Control Problem
Stochastic optimal control is used to solve optimization problems in random systems(which can be controlled) that evolve over time and whose evolution can be influencedby external forces.Optimal control theory appears in the 1950’s, when engineers became interested in prob-lems where the goal was to maximize the returns and minimize the costs of the operation,i.e. in aerospace problems a small improvement of optimal control had a large impact inreducing the costs. The solution of a stochastic optimal control problem can be foundby solving a system of di↵erential equations.In this example, based on [18], we will focus on the following dynamic stochastic di↵er-ential equation given by the controlled process:
(dX
t
= (aXt
+ bu
t
) dt+ dWt
t 2 [0, T ]
X0 = x
(1.24)
where W is a Brownian motion, Xt
, with t � 0, is called the state process taking valuesin (S,B(S)), where S is a Polish space, that is, a closed and bounded set of R.It is assumed that the system can be controlled, i.e. the system is equipped with con-trollers whose position dictates its future evolution. These controllers are characterizedby points u = (u1, ..., un) 2 Rn, the control variable.The values that can be assumed by the control variables are restricted to a certain con-trol region U 2 Rn, which is bounded.Every function u
t
defined on some interval 0 t T is called a control.
23
Definition 1.8. A continuous control ut
, defined on some time interval 0 t T , withrange in the control region U ,
u
t
2 U 8t 2 [0, T ]
is said to be an admissible control.
We consider processes (X,u) to be (Ft
)t�0-adapted and square-integrable. It is also
necessary to specify a cost function for evaluating the performance of a system quanti-tatively. We define the cost function as follows
J : U [0, T ] ! R
J(u) =1
2E
"ZT
0(|X
t
|2 + |ut
|2)dt+ |XT
|2#. (1.25)
The optimal control problem is to minimize the value of the cost function (1.25) subjectto the equation (1.24). There exists a control u 2 U that minimizes (1.25) and is uniquea.s..We suppose that u is the optimal control and X is the corresponding process state. Forany admissible controls v 2 U , we have
0 J(u+ "v)� J(u)
"
=
12E⇢R
T
0
�|Xt
|2 + |ut
+ "v
t
|2� dt+ |XT
|2�� 1
2E⇢R
T
0
�|Xt
|2 + |ut
|2� dt+ |XT
|2�
"
,
(1.26)
where X has the following dynamic:(dX
t
= aX
t
dt+ b(ut
+ "v
t
)dt+ dWt
, t 2 [0, T ],
X0 = x.
(1.27)
From (1.26) it follows:
0 12E⇢R
T
0
�|Xt
|2 � |Xt
|2 + 2ut
"v
t
+ "
2v
2t
�dt+ |X
T
|2 � |XT
|2�
"
(1.28)
=1
2E⇢Z
T
0
✓(X
t
+X
t
)(X
t
�X
t
)
"
+ 2ut
v
t
+ "v
2t
◆dt+ (X
T
+X
T
)(X
T
�X
T
)
"
�
(1.29)
We define ⇠ = lim"!0
(Xt
�X
t
)"
. From (1.24) and (1.27), it follows that ⇠ satisfies:
(d⇠
t
= (a⇠t
+ bv
t
)dt, t 2 [0, T ],
⇠0 = 0.(1.30)
24
Letting "! 0, equation (1.29) becomes:
0 E⇢Z
T
0(X
t
⇠
t
+ u
t
v
t
)dt+X
T
⇠
T
�. (1.31)
We introduce the following BSDE:
(dY
t
= �(aYt
+X
t
)dt+ Z
t
dWt
, t 2 [0, T ],
Y
T
= X
T
.
(1.32)
We suppose that the BSDE (1.32) admits a unique adapted solution (Y, Z). By applyingIto’s formula to Y
t
⇠
t
we get:
E(XT
⇠
T
) = E(YT
⇠
T
)
= EZ
T
0{(�aY
t
�X
t
)⇠t
+ Y
t
(a⇠t
+ bv
t
)} dt
= EZ
T
0{�X
t
⇠
t
+ Y
t
bv
t
}dt.
By (1.31) we get,
0 EZ
T
0(bY
t
+ u
t
)vt
dt
As v is arbitrary, we get the following
u
t
= �bY
t
a.s. 8t 2 [0, T ]. (1.33)
Now, since Y is part of the solution of the BSDE (1.32) and is Ft
-adapted, so that u isan admissible control. By substituting (1.33) in (1.24) we arrive at the following optimalsystem: 8
>>>><
>>>>:
dXt
= (aXt
� b
2Y
t
)dt+ dWt
, t 2 [0, T ],
dYt
= �(aYt
+ bX
t
)dt+ Z
t
dWt
, t 2 [0, t],
X0 = x
Y
T
= X
T
.
Finally, we have a system of one Forward Stochastic Di↵erential Equation (which areintroduced later in the thesis) as well as a Backward Stochastic Di↵erential Equation,such a system is known as a Forward Backward Stochastic Di↵erential Equation (FB-SDE). If we can solve the FBSDE and find a unique adapted solution (X,Y, Z) then u
in (1.32) is the optimal control that solves the original problem.
1.5.2. Hedging in the Black-Scholes model
We give an example for the application of BSDEs in risk hedging. In a financial marketwe have the following: for t 2 [0, T ]
25
1. There exists a risk-free asset, whose price is modelled by:(dB
t
= B
t
rdt,
B0 = y 2 R,(1.34)
where r denotes the interest rate.
2. We model a stock with a risky asset by the following forward SDE:(dS
t
= S
t
(µdt+ �dWt
),
S0 = x 2 R,(1.35)
where µ 2 R,� > 0, µ is called drift and � is the volatility.
The solutions of equations (1.34) and (1.35) are given by
B
t
= ye
rt
, (1.36)
S
t
= x exp
✓✓µ� 1
2�
2
◆t+ �W
t
◆, t � 0. (1.37)
A portfolio is a couple of adapted processes (at
, b
t
) that represents the amount of invest-ment in both assets at time t. The wealth process is given by
Y
t
= a
t
S
t
+ b
t
B
t
, t 2 [0, T ]. (1.38)
A main assumption is that Y is self-financing, which means that the wealth processsatisfies the following
dYt
= a
t
dSt
+ b
t
dBt
= a
t
S
t
(µdt+ �dWt
) + rb
t
B
t
dt.
As bt
B
t
= Y
t
� a
t
S
t
, then we have
dYt
= (rYt
+ a
t
S
t
(µ� r))dt+ a
t
S
t
�dWt
Now by putting Z
t
= a
t
S
t
�, we get
dYt
= rY
t
dt+ Z
t
µ� r
�
dt+ Z
t
dWt
.
One interesting question in the financial market is: At which price v should one sellthe European call option?
The answer is by replicating the portfolio: the seller sells the option at the price v andinvests this sum in the market. The value of his portfolio is characterized by the SDE:
(dY
t
= rY
t
dt+ µ�r
�
Z
t
dt+ Z
t
dWt
Y0 = v.
26
The problem is to find v and Z such that the solution of the previous SDE verifiesY
T
= ⇠ = (ST
�K)+,
(dY
t
= rY
t
dt+ µ�r
�
Z
t
dt+ Z
t
dWt
Y
T
= ⇠.
In this case v is called the fair price of the option and it su�ces to sell the option at theprice v = V0. We see that pricing the option is to solve a linear BSDE.
27
2. FBSDEs and Feynman-Kac formula
In this chapter we present the Forward-Backward Stochastic Di↵erential Equations (FB-SDE). FBSDEs provide an intensively studied modelling tool for stochastic control prob-lems as well as for financial mathematics. They appeared for the first time in Bismut’spaper [2] and then studied in a general way by Pardoux and Peng in [19].At the end of this chapter the Feynman-Kac representation theorem is introduced whichgives us a method to connect PDEs and stochastic processes. We can solve PDEs bymaking simulations for the stochastic process.The books [4], [20] and the article [9] are used as the main references for this chapter.
2.1. Forward-Backward Stochastic Di↵erential Equations
We consider the solution of certain BSDEs associated with some forward classical stochas-tic di↵erential equations. We are going to suppose that the randomness of the parameters(f, ⇠) of the BSDE comes from the state of the forward equation.
We consider the following Forward Stochastic Di↵erential Equation:(dX
s
= b(s,Xs
)ds+ �(s,Xs
)dWs
s 2 [t, T ],
X
t
= x,
(2.1)
where the coe�cientsb : [0, T ]⇥ R ! R,� : [0, T ]⇥ R ! R,
satisfy the following assumptions with C > 0���(t, x)� �(t, x0)
��+��b(t, x)� b(t, x0)
�� C
��x� x
0��, H1
|�(t, x)|+ |b(t, x)| C(1 + |x|). H2
Under the assumptions H1 and H2, there exists a unique solution for (2.1) (see TheoremA.12) which we denote as Xt,x
s
, for s 2 [0, T ].
Now we consider the associated BSDE,(�dY
s
= f(s,Xt,x
s
, Y
s
, Z
s
)ds� Z
s
dWs
, s 2 [0, T ],
Y
T
= g(Xt,x
T
),(2.2)
where the coe�cientsf : [0, T ]⇥ R⇥ R⇥ R ! Rg : R ! R
28
satisfy the following assumptions with C > 0 such that��f(t, x, y, z)� f(t, x0, y0, z0)
�� C(|y � y
0|+ |z � z
0|+ |x� x
0|) H3
|g(x)| C(1 + |x|). H4
From Theorem 1.5 we know that there exists a unique solution for (2.2) which we denoteas (Y t,x
s
, Z
t,x
s
) with s 2 [0, T ]. Notice that we needed to introduce a new assumption H4as the function g in the BSDE (2.2) is linked to the SDE (2.1).
We rewrite (2.1) and (2.2) in one system as follows(X
t,x
s
= x+Rs
t
b(r,Xt,x
r
)dr +Rs
t
�(r,Xt,x
r
)dWr
s � t
Y
t,x
s
= g(Xt,x
T
) +RT
s
f(r,Xt,x
r
, Y
t,x
r
, Z
t,x
r
)dr � R T
s
Z
t,x
r
dWr
s � t
(2.3)
The system (2.3) is a combination of a forward SDE and a BSDE, where the terminalcondition of the BSDE is now allowed to depend on the process X
t,x
s
. Such a type ofequation is called a Forward-Backward Stochastic Di↵erential Equation (FBSDE). Xt,x
s
represents the forward component and (Y t,x
s
, Z
t,x
s
) the backward component.
It is important to notice that the solution of the backward component does not appearin the coe�cients of the forward SDE. We call the system (2.3) a decoupled FBSDE.
A more general FBSDE has the following form(X
t
= X0 +Rt
0 b(s,Xs
, Y
s
, Z
s
)ds+Rt
0 �(s,Xs
, Y
s
)dWs
,
Y
t
= g(XT
) +RT
t
f(s,Xs
, Y
s
, Z
s
)ds� R T
t
Z
s
dWs
.
We observe that the solution of the backward component appears in the coe�cientsof the forward component, we call this class of FBSDE, coupled FBSDE. This type ofFBSDE is used in the application to carbon emissions markets. However the theory ofthe coupled BSDEs is out of scope for this thesis, for further reading we refer to thebook [4].
2.1.1. Regularity Properties of Solutions
Proposition 2.1. [9] For each t 2 [0, T ] and x 2 R, there exists C � 0 such that
E
sup
0sT
��Y
t,x
s
��2!
+ E✓Z
T
0
��Z
t,x
s
��2 ds◆
C(1 + |x|2).
Proposition 2.2. [9] For each t, t
0 2 [0, T ], x, x0 2 R and if f, g are Lipschitz in x,uniformly in t concerning f there exists C � 0 such that
E
sup
0sT
���Y t,x
s
� Y
t
0,x
0s
���2!
+ E✓Z
T
0
���Zt,x
s
� Z
t
0,x
0s
���2ds
◆
C(1 + |x|2)(|x� x
0|2 + |t� t
0|).
29
2.1.2. Markov Properties of Solutions
In this subsection, we show that the the Markov property of the solution of an SDEtranslates to the solution of the BSDE. This property is mainly used to study the linkto PDEs. We introduce the following filtration F t
s
which denotes the future �-algebra ofW after t, i.e.
F t
s
= �(Wu
�W
t
, t u s).
Proposition 2.3. [4] If (t, x) 2 [0, T ]⇥R, then⇣X
t,x
s
, Y
t,x
s
⌘, where t s T is adapted
to the filtration F t
s
. In particular Y
t,x
t
is deterministic. We can choose a version Z
t,x
s
where t s T adapted to the filtration F t
s
. Since Y
t,x
t
is deterministic we can definea function
u(t, x) := Y
t,x
t
,
for all (t, x) 2 [0, T ]⇥ R.
Proposition 2.4. [4] The function u verifies, for any (t, x), (t0, x0)
|u(t, x)� u(t0, x0)| C(|x� x
0|+ |t� t
0| 12 (1 + |x|)),
for some c � 0.
By the help of the function u we can establish the Markov property for the BSDEs.
Proposition 2.5. [4] Let t 2 [0, T ] and x be a square integrable random variable. ThenP-a.s.,
Y
t,x
s
= u(s,Xt,x
s
), 8s 2 [t, T ].
This property is called the Markov property.
2.2. The link with PDEs: Feynman-Kac formula
We introduce a semi-linear parabolic PDE, as follows:
(u
0(t, x) +Au(t, x) + f(t, x, u(t, x), Du(t, x)�(t, x)) = 0, (t, x) 2 [0, T ]⇥ R,u(T, x) = g(x),
(2.4)
where A represents the linear di↵erential operator:
Au(t, x) = b(t, x)Du(t, x) +1
2
��
2(t, x)D2u(t, x)
�. (2.5)
For what concerns the notation, when u is a function of t and x we denote u0 the partialdi↵erential in time, Du the partial di↵erential in space of first order and D
2u the partial
di↵erential in space of second order.
Feynman-Kac
30
The following proposition gives us a probabilistic representation formula for the solutionof a semi-linear parabolic PDE. Hence by solving the PDE (2.4) we can deduce thesolution of the FBSDE (2.3).
Proposition 2.6. [9] Let u 2 C2([0, T ]⇥R) and suppose that there exists a constant Csuch that, for each (t, x) 2 [0, T ]⇥ R, it holds
|u(s, x)|+ |Du(s, x)�(s, x)| C(1 + |x|).
Moreover let u be the solution of the semilinear parabolic PDE,
(u
0(t, x) +Au(t, x) + f(t, x, u(t, x), Du(t, x)�(t, x)) = 0, (t, x) 2 [0, T ]⇥ R,u(T, x) = g(x),
where A is defined as in (2.5). Then for any (t, x) 2 [0, T ]⇥ R,
Y
t,x
t
:= u(t, x),
where {(Y t,x
s
, Z
t,x
s
), t s T} is the unique solution of the BSDE in (2.3). Moreover
(Y t,x
s
, Z
t,x
s
) =⇣u(s,Xt,x
s
), Du(t,Xt,x
s
)�(t,Xt,x
s
)⌘.
Proof. We apply Ito’s lemma to u(s,Xt,x
s
), we get
du(s,Xt,x
s
) =u
0(s,Xt,x
s
)ds+Du(s,Xt,x
s
)dXt,x
s
+1
2�
2(s,Xt,x
s
)D2u(s,Xt,x
s
)ds
=
✓u
0(s,Xt,x
s
) +1
2�
2(s,Xt,x
s
)D2u(s,Xt,x
s
) +Du(s,Xt,x
s
)b(s,Xt,x
s
)
◆ds
+Du(s,Xt,x
s
)�(s,Xt,x
s
)dWs
=�u
0(s,Xt,x
s
) +Au(s,Xt,x
s
)�ds+Du(s,Xt,x
s
)�(s,Xt,x
s
)dWs
.
(by definition of A (2.5))
As u is a solution of the PDE (2.4), we have u
0 +Au = �f , we get
du(s,Xt,x
s
) = �f
�s,X
t,x
s
, u(s,Xt,x
s
), Du�(s,Xt,x
s
)�ds+Du�(s,Xt,x
s
)dWs
,
which proves the statement.
To get the reverse way, i.e. studying the BSDE and being able to deduce the con-struction of the solution of the PDE we need to define the following viscosity definition.The basic idea of viscosity solution (see for example [11]) is to replace the di↵erentialDu(x) at a point x where it does not exist (for example because of a kink in u) with thedi↵erential D�(x) of a smooth function � touching the graph of u, from above for thesub solution condition and from below for the super solution one, at the point x.
31
Definition 2.7. Suppose u 2 C([0, T ] ⇥ R,R) with u(T, x) = g(x) s.t. x 2 R, then u
is called viscosity sub solution of the PDE (2.4), if for each � 2 C
2([0, T ]⇥R), we have,
�
0(t, x) +A�(t, x) + f (t, x,�(t, x), D�(t, x)�(t, x)) � 0,
8(t, x) 2 [0, T )⇥ R whenever u� � has a local maximum.Suppose u 2 C([0, T ] ⇥ R,R) satisfies u(T, x) = g(x) s.t. x 2 R, then u is calledviscosity super solution of the PDE (2.4), if for each � 2 C
2([0, T ]⇥ R) , we have,
�
0(t, x) +A�(t, x) + f(t, x,�(t, x), D�(t, x)�(t, x)) 0,
8(t, x) 2 [0, T )⇥ R whenever u� � has a local minimum.Hence u 2 C([0, T ] ⇥ R,R) is called a viscosity solution of (2.4) if it is both a viscositysub- and super solution.
For a further reading on viscosity solution, we refer to the papers [6] and [8].
Theorem 2.8. [4] The function u(t, x) := Y
t,x
t
, as defined in (2.3), continuous on[0, T ]⇥ R, is a viscosity solution of the PDE (2.4).
Proof. By construction u is continuous and u(T, ·) = g(·) so we only need to prove is theviscosity sub solution property. Let � in C2 such that u� � has at (t0, x0) (0 < t0 < T )a local maximum equal to 0. We want to show that
�
0(t0, x0) +A�(t0, x0) + f(t0, x0, u(t0, x0), D��(t0, x0)) � 0.
We will proceed by finding a contradiction, so we assume that there exists � > 0 suchthat
�
0(t0, x0) +A�(t0, x0) + f (t0, x0, u(t0, x0), D��(t0, x0)) = �� < 0,
The function u� � has a local maximum at (t0, x0) equal to 0, this means that8(t, x) close to (t0, x0) we have,
(u� �)(t, x) (u� �)(t0, x0)
u(t, x)� �(t, x) u(t0, x0)� �(t0, x0) = 0
u(t, x) �(t, x).
By continuity there exists 0 < ↵ t�t0 such that for all t0 t t0+↵ and |x�x0| ↵,
u(t, x) �(t, x)
�
0(t, x) +A�(t, x) + f(t, x,�(t, x), D��(t, x)) ��2< 0.
Define the stopping time,
⌧ = inf{u � t0, |Xt0,x0u
� x0| � ↵} ^ (t0 + ↵).
32
Since X
t0,x0 is a continuous process, we have���Xt0,x0
⌧
� x0
��� ↵.
By Ito formula applied to �(r,Xt0,x0r
) between u^⌧ and (t0+↵)^⌧ = ⌧ for t0 u t0+↵,we obtain
�(u ^ ⌧, Xt0,x0u^⌧ ) = �(⌧, Xt0,x0
⌧
)�Z
⌧
u^⌧
��
0 +A�� (r,Xt0,x0r
)ds (2.6)
�Z
⌧
u^⌧D��
�r,X
t0,x0r
�dW
r
We consider two couples of processes.On the one hand, for t0 u t0 + ↵, we define
(Y 0u
, Z
0u
) :=⇣�(u ^ ⌧, Xt0,x0
u^⌧ ), {u⌧}D��(r,Xt0,x0r
)⌘.
By replacing Y
0u
and Z
0u
in (2.6), we get
Y
0u
= �(⌧, Xt0,x0⌧
)+
Zt0+↵
u
� {r⌧}��
0 +A�� (r,Xt0,x0r
)dr�Z
t0+↵
u
Z
0r
dWr
, t0 u t0+↵.
On the other hand, for t0 u t0 + ↵, we define
(Yu
, Z
u
) =⇣Y
t0,x0u^⌧ , {u⌧}Z
t0,x0u
⌘.
By replacing Y
u
and Z
u
in in the backward SDE of the system (2.3), we get
Y
u
= Y
t0+↵
+
Zt0+↵
u
{r⌧}f(r,Xt0,x0r
, Y
r
, Z
r
)dr �Z
t0+↵
u
Z
r
dWr
, t0 u t0 + ↵.
The Markov property (2.5) implies that P-a.s. for all t0 r t0 + ↵, Y
t0,x0r
=u(r,Xt0,x0
r
), and thus Yt0+↵
= Y
t0,x0⌧
= u(⌧, Xt0,x0⌧
) and we get
Y
u
= u(⌧, Xt0,x0⌧
) +
Zt0+↵
u
{r⌧}f�r,X
t0,x0r
, u(r,Xt0,x0r
), Zr
�dr �
Zt0+↵
u
Z
r
dWr
,
where t0 u t0 + ↵.
In the next step we will apply the Comparison Theorem (1.7) to (Y 0u
, Z
0u
) and (Yu
, Z
u
).From the definition of ⌧ we get,
u(⌧, Xt0,x0⌧
) �(⌧, Xt0,x0⌧
)
{r⌧}f(r,Xt0,x0r
, u(r,Xt0,x0r
)D��(r,Xt0,x0r
)) � {r⌧}��
0 +A�� (r,Xt0,x0r
)
Moreover, we always have by definition of ⌧ ,
E✓Z
t0+↵
t0
� {r⌧}��
0 +A�+ f
� ⇣r,X
t0,xr
, u(r,Xt),x
r
), D��(r,Xt0,xr
)⌘dr
◆
� E(⌧ � t0)�
2> 0
This quantity is indeed strictly positif, as � > 0 and ⌧ > t since���Xt0,x0
t0� x0
��� = 0 < ↵.
We can apply the strict version of the Comparison Theorem (1.7) to obtain u(t0, x0) =Y
t0 < Y
0t0= �(t0, x0). This shows the contradiction as we have u(t0, x0) = �(t0, x0).
33
3. Numerical Methods for BSDEs
Typically the dynamics of stock prices and interest rates are driven by a continuous-time stochastic process. However, simulation is done in discrete time steps. Hence, thefirst step in any simulation scheme is to find a way to ”discretize” a continuous-timeprocess into a discrete time process. We will discretize the forward component by thesimple Euler scheme and the backward component by a theta-scheme. The use of atheta-scheme introduces two parameters ✓1 and ✓2, which are used to generate multiplediscretizations.It is the Feynman–Kac theorem, proven in the previous section, that relates the con-ditional expectation of the value of a contract payo↵ function under the risk-neutralmeasure to the solution of a partial di↵erential equation.The existing numerical methods can be classified into 3 major groups:
• PDE methods
• Monte-Carlo simulation
• Numerical Integration Methods (often rely on Fourier )
Each of them have advantages and disadvantages and the last one is often used forCalibration purposes. We will use probabilistic numerical methods to solve BSDEswhich will rely on time discretization of the stochastic process and and approximationsfor the appearing conditional expectations.In this chapter we will make use of the following defined FBSDE
8><
>:
X
t
= X0 +Rt
0 b(s,Xs
)ds+Rt
0 �(s,Xs
)dWs
X0 = x0, (3.1)
Y
t
= ⇠ +
ZT
t
f(s,Xs
, Y
s
, Z
s
)ds�Z
T
t
Z
s
dWs
⇠ = g(XT
). (3.2)
The equation (3.1) represents a forward SDE and (3.2)represents a BSDE. As seen inthe previous chapter we know that this FBSDE satisfies the Markov property.Let ⇧ be a partition of time points 0 = t0 < t1 < t2 < t3 < .. < t
i
< .. < t
M
= T ,�t := t
i+1 � t
i
, be fixed time steps and �W
t
i
:= W
t
i+1 � W
t
i
, where W
t
is a Wiener
process. Notice that the increments �W
t
i
⇠ N (0,p�t).
For notational simplicity we use: X
i
= X
t
i
, Y
i
= Y
t
i
, Z
i
= Z
t
i
. To determine the valuesX
⇧i+1 for i = 0, ...,M � 1, we use three di↵erent Taylor schemes: Euler, Milstein and 2.0
weak Taylor.
34
3.1. Discretization of the SDE
In this section, based on the paper [29], we will approximate the numerical solution ofthe FSDE (3.1). Just as in the deterministic case, we can derive numerical schemesby looking at the Taylor expansions of certain functions. In the thesis we considerthe stochastic setting. First we introduce the following definitions of strong and weakconvergence.
Definition 3.1. LetX be the solution of the FSDE (3.1) andX
⇧i
be the time-discretizedapproximation of X. A time-discretized approximation X
⇧i
converges to the stochasticprocess X in the strong sense with order ↵1, if there exists a constant C 2 R such that:
E[|X⇧i
�X
i
|] C�t
↵1.
Definition 3.2. LetX be the solution of the FSDE (3.1) andX
⇧i
be the time-discretizedapproximation of X. A time-discretized approximation X
⇧i
converges to the stochasticprocess X in the weak sense, with order ↵2 if there exists a constant C 2 R such thatfor every infinitely often di↵erentiable function � : Rd ! R with at most polynomiallygrowing derivatives it holds that:
|E[�(X⇧i
)]� E[�(Xi
)]| C�t
↵2.
3.1.1. Euler-Maruyama scheme
Let k : R ! R be a function that is twice di↵erentiable and an Ito process Xt
with driftterm b(t,X
t
) 2 C2, and di↵usion term �(t,X
t
) 2 C2.Consider the forward SDE of (3.1),
X
t
= X0 +
Zt
t0
b(s,Xs
)ds+
Zt
t0
�(s,Xs
)dWs
, 8t 2 [t0, T ]. (3.3)
By Ito’s lemma we get:
k(Xt
) =k(Xt0) +
Zt
t0
b(s,X
s
)k0(Xs
) +1
2�(s,X
s
)2k00(Xs
)
�ds+
Zt
t0
�(s,Xs
)k0(Xs
)dWs
.
(3.4)
Define:
L
0k := bk
0 +1
2�
2k
00
L
1k := �k
0.
We can rewrite (3.4) as :
k(Xt
) = k(Xt0) +
Zt
t0
L
0k(X
s
)ds+
Zt
t0
L
1k(X
s
)dWs
. (3.5)
35
Consider now the integral form of Xt
in (3.3) and substitute the functions b and � fork in the equation above (3.5), we get the so called Ito-Taylor expansion of first order ofX
t
:
X
t
=X
t0 +
Zt
t0
✓b(t0, Xt0) +
Zs
t0
L
0b(t
p
, X
p
)dp+
Zs
t0
L
1b(t
p
, X
p
)dWp
◆ds
+
Zt
t0
✓�(t0, Xt0) +
Zs
t0
L
0�(t
p
, X
p
)dp+
Zs
t0
L
1�(t
p
, X
p
)dWp
◆dW
s
=X
t0 + b(t0, Xt0)
Zt
t0
ds+ �(t0, Xt0)
Zt
t0
dWs
+R
t
=X
t0 + b(t0, Xt0)(t� t0) + �(t0, Xt0)(Wt
�W
t0) +R
t
,
where Rt
is the remaining term consisting of double integrals. This Ito-Taylor expansiongives us the following Euler-Maruyama scheme as an approximation for the FSDE in(3.3) :
(X
⇧i+1 = X
⇧i
+ b(ti
, X
⇧i
)�t+ �(ti
, X
⇧i
)�W
i
8i = 0, ..,M � 1
X
⇧0 = x0.
(3.6)
Proposition 3.3. [30] Let X be given by (3.3) and assume conditions H1 and H2 (fromChapter 2) such that there exists a unique solution. Then the Euler-Maruyama scheme(3.6) has order of strong convergence ↵1 =
12 and order of weak convergence ↵2 = 1.
3.1.2. The Milstein Scheme
The Milstein approximation for the SDE in (3.3) has the form:(X
⇧0 = x0
X
⇧i+1 = X
⇧i
+ b(ti
, X
⇧i
)�t+ �(ti
, X
⇧i
)�W
i
+ 12�(ti, X
⇧i
)D�(ti
, X
⇧i
)��W
2i
��t
�,
for i = 0, ...,M � 1
Proposition 3.4. [30] Let X be given by (3.3) and assume conditions H1 and H2 suchthat there exists a unique solution. Then the Milstein scheme (??) has order of strongconvergence ↵1 = 1 and order of weak convergence ↵2 = 1.
3.1.3. The weak Taylor Scheme of order 2.0
The weak Taylor scheme of order 2.0 for the SDE in (3.3), given X
⇧i
= x, has the form:8>>>><
>>>>:
X
⇧0 = x0
X
⇧i+1 = x+ b(t
i
, x)�t+ �(ti
, x)�W
i
+ 12�(ti, x)D�(ti, x)
�(�W
i
)2 ��t
�
+Db(ti
, x)�(ti
, x)�Z
i+1 +12
�b(t
i
, x)Db(ti
, x) + 12D
2b(t
i
, x)�2(x)�(�t)2
+�b(t
i
, x)D�(ti
, x) + 12D
2�(t
i
, x)�2(x)�(�W
i
�t��Z
i+1),
for i = 0, ...,M � 1 and where �Z
i+1 =12(�W
i
�t+ ⇣
i+1(�t)3/2), ⇣
i+1 ⇠ N (0, 1/3).
36
Proposition 3.5. [30] Let X be given by (3.3) and assume conditions H1 and H2. Thenthe weak Taylor scheme of order 2.0 (3.7)has order of strong convergence ↵1 = 1 andorder of weak convergence ↵2 = 2
We observe the following,
E[�Z
i+1] = 0, Var(�Z
i+1) =1
3(�t)3 and Cov(�W
i
,�Z
i+1) =1
2(�t)2/. (3.7)
If we replace �Z
i+1 by �Z
i+1 =1
2�W
i
�t as suggested in [27], then
E[�Z
i+1] = 0, Var(�Z
i+1) =1
4(�t)3 and Cov(�W
i
,�Z
i+1) =1
2(�t)2. (3.8)
This replacement has the same moments in first order and simplifies the scheme.
Remark 3.6. Similarly as in [29], the Taylor discretization scheme can be written in ageneral form, given X
⇧i
= x, as follows:
(X
⇧i+1 = x+m(t
i
, x)�t+ s(ti
, x)�W
i
+ t(ti
, x)(�W
i
)2, for i = 0, ..M � 1
X
⇧0 = x0.
(3.9)
• For the Euler scheme, we find:
8><
>:
m(ti
, x) = b(ti
, x),
s(ti
, x) = �(ti
, x),
t(ti
, x) = 0.
• For the Milstein scheme, we have:
8><
>:
m(ti
, x) = b(ti
, x)� 12�(ti, x)D�(ti, x),
s(ti
, x) = �(ti
, x),
t(ti
, x) = 12�(ti, x)D�(ti, x).
• For the 2.0-weak-Taylor scheme, we see that:
8>>>>>><
>>>>>>:
m(ti
, x) = b(ti
, x)� 12�(ti, x)D�(ti, x) +
12
�b(t
i
, x)Db(ti
, x)
+12D
2b(t
i
, x)�2(ti
, x)��t,
s(ti
, x) = �(ti
, x) + 12
�Dµ(t
i
, x)�(ti
, x) + µ(ti
, x)Dµ(ti
, x)
+12D
2µ(t
i
, x)�2(ti
, x)��t,
t(ti
, x) = 12�(ti, x)D�(ti, x).
For the discretization schemes above we can determine a characteristic function ofX
⇧i+1 (3.9), which is given in Lemma 3.7. For notational simplicity, in Lemma 3.7, we
will use that the variables only depend on the space term, i.e. m(ti
, x) = m(x), s(ti
, x) =s(x), t(t
i
, x) = t(x).
37
Lemma 3.7. [30] The characteristic function of X⇧i+1 (3.9), given X
⇧i
= x, is given by
�
x
X
⇧i+1
(u) =E[exp(iuX⇧i+1)|X⇧
i
= x]
=exp
iux+ ium(x)�t�
12u
2s
2(x)�t
1� 2iut(x)�t
!(1� 2iut(x)�t)�
12.
Proof. First we show the result when t(x) = 0,
�
x
X
⇧i+1
(u) =E⇥exp(iuX⇧
i+1)|X⇧i
= x
⇤
=E⇥exp(iux+ ium(x)�t+ ius(x)�W
i
)|X⇧i
= x
⇤
=exp (iux+ ium(x)�t)E[exp(ius(x)�W
i
)],
where �W
i
⇠ N (0,�t), we get
�
x
X
⇧i+1
(u) = exp(iux+ ium(x)�t)�N (0,�t)(us(x))
=exp
✓iux+ ium(x)�t� 1
2u
2s
2(x)�t
◆.
For t(x) 6= 0 we find that
�
x
X
⇧i+1
(u) =E⇥exp(iuX⇧
i+1)|X⇧i
= x
⇤
=E⇥exp
�iux+ ium(x)�t+ ius(x)�W
i
+ iut(x)(�W
i
)2�⇤
=E
"exp
iux+ ium(x)�t+ iut(x)
✓�W
i
+1
2
s(x)
t(x)
◆2
� iu
1
4
s
2(x)
t(x)
!#
=exp⇣iux+ ium(x)�t� iu
1
4
s
2(x)
t(x)
⌘E
"exp
t(x)iu
✓�W
i
+1
2
s(x)
t(x)
◆2!#
,
where �W
i
+ 12s(x)t(x) ⇠ N �12 s(x)
t(x) ,�t
�by (A.11) this is equivalent to
1�t
⇣�W
i
+ 12s(x)t(x)
⌘2 ⇠ �
021
⇣14
s
2(x)t
2(x)�t
⌘which denotes the chi-squared distribution with one
degree of freedom and noncenltrality parameter 14
s
2(x)t
2(x)�t
. Hence,
�
x
X
⇧i+1
(u) =exp
✓iux+ ium(x)�t+ iu
1
4
s
2(x)
t(x)
◆�
�
021
⇣12
s
2(x)
t
2(x)�t
⌘(ut(x)�t)
=exp
✓iux+ ium(x)�t+ iu
1
4
s
2(x)
t(x)
◆
exp
✓1
4
s
2(x)
t(x)
iu
1� 2iut(x)�t
◆(1� 2iut(x)�t)�
12
=exp
iux+ ium(x)�t�
12u
2s
2(x)�t
1� 2iut(x)�t
!(1� 2iut(x)�t)�
12
and the statement is proved.
38
3.2. Discretization of the BSDE
In this section we will develop a discretization scheme for the BSDE (3.1). First, theterminal condition Y
M
will be approximated by using the Euler scheme (3.6)for Xt
:
Y
M
= g(X⇧M
).
From the Feynman-Kac formula, i.e. Proposition 2.6, we know that the solution of theFBSDE (3.1)
(Yt
, Z
t
) = (u(t,Xt
),�(t,Xt
)Du(t,Xt
)).
At time T we know that u(tM
, X
M
) = g(XM
), so we have Z
M
= �(tM
, X
M
)Dg(XM
).Hence the terminal condition for Z
t
is obtained by substituting for XT
its Euler scheme:
Z
M
= �(tM
, X
⇧M
)Dg(X⇧M
).
To develop a numerical scheme for the BSDE, we focus on the discrete version of theBSDE on the interval [t
i
, t
i+1]:
Y
i
= Y
i+1 +
Zt
i+1
t
i
f(s,Xs
, Y
s
, Z
s
)ds�Z
t
i+1
t
i
Z
s
dWs
. (3.10)
By a basic Euler discretization, backward in time, the unknown value Yi+1 is required to
approximate Yi
. However, by taking the adaptability constraints on Y and Z and takingthe conditional expectation on both sides with respect to the filtration F
t
i
into account,it is possible to get a backward induction scheme. Hence, as Y
i
and Z
i
are adapted toF
t
i
, we haveE[Z
i
|Ft
i
] = Z
i
,
E[Yi
|Ft
i
] = Y
i
.
We consider (3.10) with the conditional expectation w.r.t. the filtration Ft
i
we get,
Y
i
= E[Yi+1|Ft
i
] + Eh Z
t
i+1
t
i
f(s,Xs
, Y
s
, Z
s
)ds���F
t
i
i� E
h Zt
i+1
t
i
Z
s
dWs
���Ft
i
i
= E[Yi+1|Ft
i
] +
Zt
i+1
t
i
E[f(s,Xs
, Y
s
, Z
s
)|Ft
i
]ds� Eh Z
t
i+1
t
i
Z
s
dWs
���Ft
i
i
(by Fubini’s theorem)
Since we assumed that Zt
2 H2T
, we get that its integral form is a martingale. Hence weget,
Y
i
= E[Yi+1|Ft
i
] +
Zt
i+1
t
i
E[f(s,Xs
, Y
s
, Z
s
)|Ft
i
]ds. (3.11)
39
Now we will use the theta-time discretization method [13]: a convex combination of anexplicit term, i.e. at time t
i+1 and an implicit term, i.e. at time t
i
to approximate theintegral. The equation (3.11) becomes:
Y
i
⇡ E[Yi+1|Ft
i
] +�t✓1E[f(ti, Xi
, Y
i
, Z
i
)|Ft
i
] +�t(1� ✓1)E[f(ti+1, Xi+1Yi+1, Zi+1)|Ft
i
]
(since Y
t
, Z
t
areFt
i
measurable)
= E[Yi+1|Ft
i
] +�t✓1f(ti, Xi
, Y
i
, Z
i
) +�t(1� ✓1)E[f(ti+1, Xi+1Yi+1, Zi+1)|Ft
i
],(3.12)
with ✓1 2 [0, 1].To get a numerical scheme for the process Z
i
, we multiply both sides of equation (3.10)by �W
i
and then take conditional expectations w.r.t the filtration Ft
i
. We get thefollowing
0 =E[�W
i
Y
i+1|Ft
i
] + Eh Z
t
i+1
t
i
�W
i
f(s,Xs
, Y
s
, Z
s
)ds|Ft
i
i� E
h�W
i
Zt
i+1
t
i
Z
s
dWs
���Ft
i
i
=E[�W
i
Y
i+1|Ft
i
] +
Zt
i+1
t
i
Ehf(s,X
s
, Y
s
, Z
s
)(Wi+1 �W
s
+W
s
�W
i
)���F
t
i
ids
� Eh Z
t
i+1
t
i
dWs
Zt
i+1
t
i
Z
s
dWs
���Ft
i
i
=E[�W
i
Y
i+1|Ft
i
] +
Zt
i+1
t
i
Ehf(s,X
s
, Y
s
, Z
s
)(Wi+1 �W
s
+W
s
�W
i
)���F
t
i
ids
�Z
t
i+1
t
i
E[Zs
|Ft
i
]ds
=E[�W
i
Y
i+1|Ft
i
] +
Zt
i+1
t
i
E[f(s,Xs
, Y
s
, Z
s
)(Wi+1 �W
s
)|Ft
i
]ds
+
Zt
i+1
t
i
E[f(s,Xs
, Y
s
, Z
s
)(Ws
�W
i
)|Ft
i
]ds�Z
t
i+1
t
i
E[Zs
|Ft
i
]ds
=E[�W
i
Y
i+1|Ft
i
] +
Zt
i+1
t
i
E[E[f(s,Xs
Y
s
, Z
s
)(Wi+1 �W
s
)|Ft
s
]|Ft
i
]ds
+
Zt
i+1
t
i
E[f(s,Xs
, Y
s
, Z
s
)(Ws
�W
i
)|Ft
i
]ds�Z
t
i+1
t
i
E[Zs
|Ft
i
]ds
=E[�W
i
Y
i+1|Ft
i
] +
Zt
i+1
t
i
E[f(s,Xs
, Y
s
, Z
s
)E[(Wi+1 �W
s
)|Ft
s
]|Ft
i
]ds
+
Zt
i+1
t
i
E[f(s,Xs
, Y
s
, Z
s
)(Ws
�W
i
)|Ft
i
]ds�Z
t
i+1
t
i
E[Zs
|Ft
i
]ds
=E[�W
i
Y
i+1|Ft
i
] +
Zt
i+1
t
i
E[f(s,Xs
, Y
s
, Z
s
)(Ws
�W
i
)|Ft
i
]ds�Z
t
i+1
t
i
E[Zs
|Ft
i
]ds,
(3.13)
40
where we used Fubini’s theorem to switch the order of integration in the first integral.By theta-approximation of both integrals in (3.13), we get:
0 ⇡E[�W
i
Y
i+1|Ft
i
] +�t(1� ✓2)E[f(ti+1, Xi+1, Yi+1, Zi+1)�W
i
|Ft
i
]
��t✓2Zi
��t(1� ✓2)E[Zi+1|Ft
i
], ✓2 2 [0, 1], (3.14)
where we used the Ito-adaptedness property of Yi
and Z
i
and the fact that �W
i
|Ft
i
⇠N (0,�t). The above equations (3.12) and (3.14) lead to the following discrete-timeapproximation (Y ⇧
, Z
⇧) for (Y, Z):
8>>>>>>>>><
>>>>>>>>>:
Y
⇧M
= g(X⇧M
),
Z
⇧M
= �Dg(X⇧M
),
Z
⇧i
= �✓�12 (1� ✓2)E[Z⇧
i+1|Ft
i
] + 1�t
✓
�12 E[Y ⇧
i+1�W
i
|Ft
i
]
+✓�12 (1� ✓2)E[f(ti+1, X
⇧i+1, Y
⇧i+1, Z
⇧i+1)�W
i
|Ft
i
],
Y
⇧i
= E[Y ⇧i+1|Ft
i
] +�t✓1f(ti, X⇧i
, Y
⇧i
, Z
⇧i
)
+�t(1� ✓1)E[f(ti+1, X⇧i+1, Y
⇧i+1, Z
⇧i+1)|Ft
i
].
We observe that Y ⇧i
and Z
⇧i
depend on the value X
⇧i
, so when X
⇧i
= x, then
8>>>>>>>>>><
>>>>>>>>>>:
Y
⇧M
= g(X⇧M
), (3.15)
Z
⇧M
= �Dg(X⇧M
), (3.16)
Z
⇧i
= �✓�12 (1� ✓2)E[Z⇧
i+1
�X
⇧i+1
� |X⇧i
= x] (3.17)
+1
�t
✓
�12 E[Y ⇧
i+1
�X
⇧i+1
��W
i
|X⇧i
= x],
Y
⇧i
= E[Y ⇧i+1
�X
⇧i+1
� |X⇧i
= x] +�t✓1f(ti, x, Y⇧i
(x) , Z⇧i
(x)) (3.18)
+�t(1� ✓1)E[f(ti+1, X⇧i+1, Y
⇧i+1(X
⇧i+1), Z
⇧i+1(X
⇧i+1))|X⇧
i
= x],
for i = M � 1, ..., 0.
41
4. The Fourier-Cosine method forBackward Stochastic Di↵erentialEquations
In this chapter we present an option pricing method for European options with oneunderlying asset, based on the Fourier-cosine series. This method developed by Fangand Oosterlee [12] is called the one-dimensional COS method and as it is used for solvingBSDEs a di↵erent name is BCOS method. The key idea of this method is the relationof the characteristic function with the series coe�cients of the Fourier-cosine expansionof the discounted expected payo↵.This method covers a range of application in di↵erent underlying dynamics, includingLevy processes and the Heston stochastic volatility model, and various types of optioncontracts. In this thesis we will only deal with European options in particular. Thischapter is based on [26] [27]. To get a better understanding, the theory of Fourier seriesis introduced in the first section which is based on the book [7].
4.1. Fourier Transform & Fourier Cosine series
Definition 4.1. Let p(x) be a piecewise continuous real function over R which satisfiesthe integrability condition Z +1
�1|p(x)|dx < 1.
The Fourier transform of p(x) is defined by
p(!) =
Z +1
�1e
i!x
p(x)dx, x 2 R. (4.1)
The inverse Fourier transform of p(x) is given by
p(x) =1
2⇡
Z +1
�1e
�iyx
p(y)dy, y 2 R. (4.2)
Definition 4.2. Let R be a random variable having probability density function f(x),such that Z +1
�1f(x)dx = 1.
42
The Fourier transform of f(x) is the characteristic function, denoted by � such that,
�(u) = Ehe
iuR
i=
Z +1
�1e
iux
f(x)dx, u 2 R. (4.3)
The probability density function is the inverse Fourier transform of the characteristicfunction. The characteristic function and probability density function form a Fourierpair.
A Fourier series is an expansion of a periodic function f(x) in terms of an infinitesum of sines and cosines. A function f(x) supported on the domain x 2 [�⇡,⇡] can bewritten as its Fourier series by
f(x) =1
2A0 +
1X
n=1
A
n
cos(nx) +1X
n=1
B
n
sin(nx),
A0 :=1
⇡
Z⇡
�⇡
f(y)dy,
A
n
:=1
⇡
Z⇡
�⇡
f(y) cos(ny)dy,
B
n
:=1
⇡
Z⇡
�⇡
f(y) sin(ny)dy,
For a symmetric function, i.e. f(x) = f(�x) 8x, the coe�cients Bn
will be zero. TheCOS method is based on the Fourier cosine series.
A function f(x) supported on the domain x 2 [a, b] can be written as its Fourier cosineseries by
f(x) =1
2A0 +
1X
k=1
A
k
cos
✓k⇡
x� a
b� a
◆
=1X
k=0
0A
k
cos
✓k⇡
x� a
b� a
◆,
whereP1
k=00 indicates that the first term in the summation is weighted by 1
2 and A
k
isdefined by
A
k
:=2
b� a
Zb
a
f(y) cos
✓k⇡
y � a
b� a
◆dy. (4.4)
For functions supported on any other finite interval, say [a, b] 2 R, the Fourier-cosineseries expansion can easily be obtained via a change of variables. The coe�cients A
k
can be approximated using the Fourier transform of f(y) as we explain below. Noticethat a cosine-function can be written as follows
cos
✓k⇡
y � a
b� a
◆= R
⇢exp
✓k⇡
y � a
b� a
◆�.
43
where R{·} represents the real part of the argument. Putting the above equality into(4.4) gives
A
k
=2
b� a
Zb
a
f(y)R⇢exp
✓k⇡
y � a
b� a
◆�dy
A
k
=2
b� a
R⇢Z
b
a
f(y) exp
✓k⇡
y � a
b� a
◆dy
�.
Suppose the integral over the whole domain is a good approximation of the integral ofthe interval. The coe↵cients A
k
can be written as
A
k
=2
b� a
R⇢Z
b
a
f(y) exp
✓i
✓k⇡
b� a
◆y � ik⇡a
b� a
◆dy
�
⇡ 2
b� a
R⇢Z
Rf(y) exp
✓i
✓k⇡
b� a
◆y � ik⇡a
b� a
◆dy
�
⇡ 2
b� a
R⇢�
✓k⇡
b� a
◆exp
✓�i
k⇡a
b� a
◆�.
A method for pricing European options with numerical integration techniques is therisk-neutral valuation formula:
v(St
, t) = e
�r(T�t)EQ[v(ST
, T )|St
] = e
�r(T�t)Z
Rv(S
T
, T )f(ST
|St
)dST
, (4.5)
where v denotes the option value, t is the initial time T is the marturity time, EQ[·] isthe expectation under the risk-neutral measure Q, S
t
and S
T
are the price variables ofthe underlying asset at time t and T , f(S
T
|St
) is the density function of ST
given S
t
, ris the risk-neutral interest rate.We insert in (4.5):
x := S
t
, y := S
T
, �t := T � t
v(x, t) = e
�r�tEQ[v(y, T )|x] = e
�r�t
Z
Rv(y, T )f(y|x)dy. (4.6)
The integral (4.6) will be approximated with the COS method in 5 steps
1. Truncate the integration part.The density function f(y|x) in (4.6) decays to zero very fast as y ! ±1. Therefore,v(x, t) can be approximated by some finite integration range [a, b] ⇢ R
v(x, t) ⇡ v1(x, t) = e
�r�t
Zb
a
v(y, T )f(y|x)dy.
2. Replace the density function by its cosine expansion.The density function is usually not known, so we replace the density function byits cosine expansion in y.
f(y|x) =1X
k=0
0A
k
(x) · cos✓k⇡
y � a
b� a
◆, (4.7)
44
where the series coe�cients Ak
(x) are defined as
A
k
(x) :=2
b� a
Zb
a
f(y|x) cos✓k⇡
y � a
b� a
◆dy.
Inserting (4.7) into v1(x, t), we get
v1(x, t) = e
�r�t
Zb
a
v(y, T )1X
k=0
0A
k
(x) · cos✓k⇡
y � a
b� a
◆dy. (4.8)
3. Interchange summation and integration.We interchange summation and integration in (4.8)
v1(x, t) = e
�r�t
1X
k=0
0 b� a
2A
k
(x)2
b� a
Zb
a
v(y, T ) · cos✓k⇡
y � a
b� a
◆dy (4.9)
and we define the Fourier-cosine series coe�cients Vk
of v(y, T ) on [a, b] as
V
k
:=2
b� a
Zb
a
v(y, T ) cos
✓k⇡
y � a
b� a
◆dy.
Insert Vk
into (4.9), we get
v1(x, t) = e
�r�t
1X
k=0
0 b� a
2A
k
(x)Vk
. (4.10)
The integral of the product of f(y|x) and v(y, T ) is now written as a sum of theproduct of their Fourier-cosine coe�cients.
4. Truncate the series summation.As the coe�cients A
k
, V
k
have a fast decay to zero, the summation can be trun-cated,
v2(x, t) = e
�r�t
N�1X
k=0
0 b� a
2A
k
(x)Vk
. (4.11)
5. Insert the characteristic function.We define the characteristic function of f(y|x) on the interval [a, b] by �
A
. Whetherthe characteristic function of f(y|x) is known or not known, it will always be definedon the whole domain R. The function f(y|x) decays to zero outside the domain[a, b]. Therefore �
A
does not di↵er much from �.
A
k
(x) =2
b� a
Zb
a
f(y|x) cos✓k⇡
y � a
b� a
◆dy (4.12)
=2
b� a
R(�
A
k⇡
b� a
!exp
� i
ka⇡
b� a
!)
⇡ 2
b� a
R(�
k⇡
b� a
!exp
� i
ka⇡
b� a
!).
45
We introduce F
k
(x) defined as
F
k
(x) := R(�
k⇡
b� a
!exp
� i
ka⇡
b� a
!). (4.13)
We insert Fk
(x) into (4.11) which gives the COS pricing formula:
v(x, t) ⇡ v3(x, t) = e
�r�t
N�1X
k=0
0F
k
(x)Vk
. (4.14)
4.2. Approximations of the conditional expectations
We wish to approximate the following conditional expectations:
E[Z⇧i+1
�X
⇧i+1
� |Ft
i
]
E[Y ⇧i+1
�X
⇧i+1
��W
i
|Ft
i
]
E[f(ti+1, X
⇧i+1, Y
⇧i+1
�X
⇧i+1
�, Z
⇧i+1
�X
⇧i+1
�)�W
i
|Ft
i
]
E[Y ⇧i+1
�X
⇧i+1
� |Ft
i
]
E[f(ti+1, X
⇧i+1, Y
⇧i+1(X
⇧i+1), Z
⇧i+1(X
⇧i+1))|Ft
i
]
The above conditional expectations can be categorized in two groups, with v a generalfunction:
E⇥v(t
i+1, X⇧i+1)|X⇧
i
= x
⇤(4.15)
E⇥v(t
i+1, X⇧i+1)�W
i
|X⇧i
= x
⇤. (4.16)
First we will approximate (4.15). By the Fourier-Cosine method on v(ti+1, X
⇧i+1), we
get
v(ti+1, X
⇧i+1) =
1X
k=0
0Vk
cos
k⇡
X
⇧i+1 � a
b� a
!
⇡N�1X
k=0
0Vk
cos
k⇡
X
⇧i+1 � a
b� a
!.
This gives us the following approximation for (4.15)
E⇥v(t
i+1, X⇧i+1)|X⇧
i
= x
⇤ ⇡N�1X
k=0
0Vk
E
"cos
k⇡
X
⇧i+1 � a
b� a
!����X⇧i
= x
#(4.17)
with Vk
(ti+1) =
2
b� a
Zb
a
v(ti+1, x) cos
✓k⇡
x� a
b� a
◆dx. (4.18)
46
Now we will rewrite the expectation in (4.17)
E
"cos
k⇡
X
⇧i+1 � a
b� a
!����X⇧i
= x
#
= R(E
"exp
ik⇡
X
⇧i+1 � a
b� a
!����X⇧i
= x
#)
= R(exp
✓ik⇡
�a
b� a
◆E
"exp
ik⇡
X
⇧i+1
b� a
!����X⇧i
= x
#)
= R⇢exp
✓ik⇡
�a
b� a
◆�
x
X
⇧i+1
✓k⇡
b� a
◆�. (by Lemma 3.7)(?)
We will insert the equation (?) back to (4.17) and finally for (4.15) we get
E⇥v(t
i+1, X⇧i+1)|X⇧
i
= x
⇤ ⇡N�1X
k=0
0Vk
(ti+1)R
⇢�
x
X
⇧i+1
✓k⇡
b� a
◆exp
✓ik⇡
�a
b� a
◆�.
Now we will approximate (4.16) in the same manner as (4.15) but in a slightly morecomplicated setting. We again use the Fourier-Cosine method on v(t
i+1, ⇠) and get
v(ti+1, X
⇧i+1) =
1X
k=0
0Vk
cos
k⇡
X
⇧i+1 � a
b� a
!
⇡N�1X
k=0
0Vk
cos
k⇡
X
⇧i+1 � a
b� a
!.
This gives us the following approximation for (4.16),
E⇥v(t
i+1, X⇧i+1)�W
i
|X⇧i
= x
⇤ ⇡N�1X
k=0
0Vk
E
"cos
k⇡
X
⇧i+1 � a
b� a
!�W
i
����X⇧i
= x
#(4.19)
with Vk
(ti+1) =
x
b� a
Zb
a
v(ti+1, x) cos
✓k⇡
x� a
b� a
◆dx.
From (4.19) we rewrite the expectation,
E
"cos
k⇡
X
⇧i+1 � a
b� a
!�W
i
����X⇧i
= x
#
= R(E
"exp
ik⇡
X
⇧i+1 � a
b� a
!�W
i
����X⇧i
= x
#)
= R(E
"exp
ik⇡
X
⇧i+1
b� a
!�W
i
exp
✓ik⇡
�a
b� a
◆ ����X⇧i
= x
#)
= R(exp
✓ik⇡
�a
b� a
◆E
"�W
i
exp
ik⇡
X
⇧i+1
b� a
!����X⇧i
= x
#). (4.20)
47
Applying integration by parts and substituting u = k⇡
b�a
for the expectation term inequation (4.20), we get
E�W
i
exp�iuX
⇧i+1
� ����X⇧i
= x
�=E
"�W
i
exp⇣iux+ iub(t
i
, x, y(ti
, x), z(ti
, x)�W
i
)�t
+ iu�(ti
, x, y(ti
, x), z(ti
, x))⌘����X
⇧i
= x
#(?)
by using b(ti
, x, y(ti
, x), z(ti
, x)) := m(ti
, x), �(ti
, x, y(ti
, x), z(ti
, x)) := s(ti
, x) and�W
i
⇠ N (0,�t)
(?) =1p
2⇡p�t
Z +1
�1exp (iux+ ium(t
i
, x)�t+ ius(ti
, x)⇠) ⇠ exp
✓�1
2
⇠
2
�t
◆d⇠
=� 1p2⇡
p�t
exp (iux+ ium(ti
, x)�t+ ius(ti
, x)⇠)�t exp
✓�1
2
⇠
2
�t
◆ �����
+1
�1
+1p
2⇡p�t
Z +1
�1ius(t
i
, x)�t exp (iux+ ium(ti
, x)�t+ ius(ti
, x)⇠) exp
✓�1
2
⇠
2
�t
◆d⇠
=0 + ius(ti
, x)
p�tp2⇡
Z +1
�1exp (iux+ ium(t
i
, x)�t+ ius(ti
, x)⇠) exp
✓�1
2
⇠
2
�t
◆d⇠
=ius(ti
, x)�t
p�tp2⇡
Z +1
�1
1p2⇡
p�t
exp (iu(x+m(ti
, x)�t+ s(ti
, x)⇠)) exp
✓�1
2
⇠
2
�t
◆d⇠
=ius(ti
, x)�tE [exp (iu(x+m(ti
, x)�t+ s(ti
, x)�W
i
))]
=s(ti
, x)�tEiu exp
�iu(X⇧
i
+m(ti
, x)�t+ s(ti
, x)�W
i
)� ����X
⇧i
= x
�
=s(ti
, x)�tEiu exp
�iuX
⇧i+1
� ����X⇧i
= x
�
=�(ti
, x, y(ti
, x), z(ti
, x))�tE⇥D exp(iuX⇧
i+1)|X⇧i
= x
⇤
=�(ti
, x, y(ti
, x), z(ti
, x))�tiu�
x
X
⇧i+1
(u) . (by Lemma 3.7)
Finally we get,
E⇥v(t
i+1, X⇧i+1)�W
i
|X⇧i
= x
⇤
⇡ �t�(ti
, x, y(ti
, x), z(ti
, x))N�1X
k=0
0Vk
(ti+1)R
⇢ED exp(iuX⇧
i+1)
����X⇧i
= x
�✓ik⇡
�a
b� a
◆�
= �t�(ti
, x, y(ti
, x), z(ti
, x))N�1X
k=0
0Vk
(ti+1)R
⇢i
k⇡
b� a
�
x
X
⇧i+1
✓k⇡
b� a
◆exp
✓ik⇡
�a
b� a
◆�
with Vk
(ti+1) =
2
b� a
Zb
a
v(ti+1, x) cos
✓k⇡
x� a
b� a
◆dx.
48
4.2.1. COS approximation of the function Z⇧i
For the computation of Z⇧i
in (3.17), we need to compute three expectations,E⇥Z
⇧i+1
�X
⇧i+1
� |Ft
i
⇤,E[Y ⇧
i+1
�X
⇧i+1
��W
i
|Ft
i
] andE[f(t
i+1, X⇧i+1, Y
⇧i+1
�X
⇧i+1
�, Z
⇧i+1
�X
⇧i+1
�)�W
i
|Ft
i
]. With the help of COS formulas wecan derive the following approximations for these expectations:
E⇥Z
⇧i+1
�X
⇧i+1
� |Ft
i
⇤ ⇡N�1X
k=0
0Zk
(ti+1)R
⇢�
x
X
⇧i+1
✓k⇡
b� a
◆exp
✓ik⇡
�a
b� a
◆�
E[Y ⇧i+1
�X
⇧i+1
��W
i
|Ft
i
]
⇡ �t�(ti
, x, y(ti
, x), z(ti
, x))N�1X
k=0
0Yk
(ti+1)R
⇢i
k⇡
b� a
�
x
X
⇧i+1
✓k⇡
b� a
◆exp
✓ik⇡
�a
b� a
◆�
E[f(ti+1, X
⇧i+1, Y
⇧i+1
�X
⇧i+1
�, Z
⇧i+1
�X
⇧i+1
�)�W
i
|Ft
i
]
⇡ �t�(ti
, x, y(ti
, x), z(ti
, x))N�1X
k=0
0Fk
(ti+1)R
⇢i
k⇡
b� a
�
x
X
⇧i+1
✓k⇡
b� a
◆exp
✓ik⇡
�a
b� a
◆�
By using the above equalities we obtain the COS approximation for Z⇧i
in (3.17).
4.2.2. COS approximation of the function Y ⇧i
For the computation of Y ⇧i
in (3.18), we need to compute two expectations,E⇥Y
⇧i+1
�X
⇧i+1
� |Ft
i
⇤and E[f(t
i+1, X⇧i+1, Y
⇧i+1
�X
⇧i+1
�, Z
⇧i+1
�X
⇧i+1
�)|F
t
i
], that are approx-imated by the following COS formulas:
E[Y ⇧i+1
�X
⇧i+1
� |Ft
i
] ⇡N�1X
k=0
0Yk
(ti+1)R
⇢�
x
X
⇧i+1
✓k⇡
b� a
◆exp
✓ik⇡
�a
b� a
◆�
E[f(ti+1, X
⇧i+1, Y
⇧i+1(X
⇧i+1), Z
⇧i+1(X
⇧i+1))|Ft
i
]
⇡N�1X
k=0
0Fk
(ti+1)R
⇢�
x
X
⇧i+1
✓k⇡
b� a
◆exp
✓ik⇡
�a
b� a
◆�.
When ✓1 > 0 we obtain an implicit part in (3.18), for which we perform P Picard itera-
tions [26] starting with initial guess EY
⇧i+1
�X
⇧i+1
� ����X⇧i
= x
�.
By using the above equalities and the Picard iteration method we obtain the COS ap-proximation for Y ⇧
i
in (3.18).
49
4.2.3. Fourier-cosine coe�cients
Let Yk
(ti
) be the Fourier-cosine coe�cients of Y ⇧i
(x), i.e.,
Yk
(ti
) =2
b� a
Zb
a
Y
⇧i
(x) cos
✓k⇡
x� a
b� a
◆dx, (4.21)
and Zk
(ti+1) be the Fourier-cosine coe�cients of Z⇧
i
(x), i.e.,
Zk
(ti
) =2
b� a
Zb
a
Y
⇧i
(x) cos
✓k⇡
x� a
b� a
◆dx, (4.22)
and Fk
(ti
) the Fourier-cosine coe�cients of driver f(ti
, x, Y
⇧i
(x), Z⇧i
(x)),
Fk
(ti
) =2
b� a
Zb
a
f(ti
, x, Y
⇧i
(x), Z⇧i
(x)) cos
✓k⇡
x� a
b� a
◆dx. (4.23)
At maturity T , we have the following Fourier-cosine coe�cients
Yk
(tM
) =2
b� a
Zb
a
g(x) cos
✓k⇡
x� a
b� a
◆dx, (4.24)
Zk
(tM
) =2
b� a
Zb
a
�(x)Dg(x) cos
✓k⇡
x� a
b� a
◆dx. (4.25)
4.2.4. Discretization of the Fourier-cosine coe�cient
In this subsection we will refer to the general Fourier-cosine coe�cient Vk
(ti+1) in
(4.18), but the same holds for Fk
(ti+1),Z
k
(ti+1),Y
k
(ti+1). The Fourier-cosine coe�-
cient Vk
(ti+1) of v at time points t
i
where i = 0, ...,M and j = 0, ..., N � 1 is not alwaysavailable to us. So the integral needs to be approximated with a discrete Fourier-cosinetransform. To obtain such an approximation, we take N equidistant grid-points x
n
onthe grid [a, b] and compute the value of v(t
i+1, x) on these grid-points.
Vk
(ti+1) =
2
b� a
Zb
a
v(ti+1, x) cos
✓k⇡
x� a
b� a
◆dx.
⇡ 2
b� a
N�1X
n=0
0v(t
i+1, xn) cos
✓k⇡
x
n
� a
b� a
◆�x
=2
N
N�1X
n=0
0v(t
i+1, xn) cos
✓k⇡
2n
+ 1
2N
◆. (4.26)
The approximation (4.26) is known as the discrete Fourier-cosine transform.
50
4.3. BCOS summarized
BCOS methodInitial step:Compute, or approximate, the terminal coe�cients Y
k
(tM
),Zk
(tM
) with formulas (4.24)and (4.25).Loop step: For i = M � 1 to i = 1approximate the necessary conditional expectations with Section 4.3 and compute thefunctions Y
⇧i
(x), Z⇧i
(x) and f(ti
, x, Y
⇧i
(x), Z⇧i
(x)) for x 2 [a, b] with formulas (3.17)and (3.18). Determine the corresponding Fourier-cosine coe�cients Z
k
(ti
),Fk
(ti
),Yk
(ti
)with formulas (4.21)-(4.23). Those integrals can be approximated by the discreteFourier-cosine transform, as in section 4.3.4.Terminal step:Compute Y
⇧0 (X0) and Z
⇧0 (X0).
51
5. Numerical experiment: Hedging inthe Black-Scholes model
We use the COS method for performing a numerical example taken from [26]. Thecalculations are performed in MATLAB.We compute the price v(t, S
t
) of a call option by a BSDE where the underlying assetfollows a geometric Brownian motion,
dSt
= µS
t
dt+ �S
t
dWt
.
The exact solution can be computed analytically by solving the Black-Scholes equationwith the help of the Feynman-Kac theorem for linear PDEs.For the derivation of the Black-Scholes PDE we set up a self-financing portfolio Y
t
witha
t
assets and Y
t
� a
t
S
t
bonds as well as r the risk-free return.In this model we assume that the market is complete, there are no trading restrictionsand the option can be exactly replicated by the hedging portfolio Y
T
= max(ST
�K, 0).Then, the option value at initial time should be equal to the initial value of the portfolio.The portfolio follows the following SDE
dYt
=r(Yt
� a
t
S
t
)dt+ a
t
dSt
=
✓rY
t
+µ� r
�
�a
t
S
t
◆dt+ �a
t
S
t
dWt
.
If we set Zt
= �a
t
S
t
, then (Y, Z) solves the BSDE
dY
t
= �f(t, Yt
, Z
t
)dt+ Z
t
dW
t
f(t, Yt
, Z
t
) = �rY
t
� µ� r
�
Z
t
Y
T
= max(ST
�K, 0).
Y
t
corresponds to the value of the portfolio and Z
t
is related to the hedging strategy.In this case, the option value is given by v(t, S
t
) = Y
t
and �St
Dv(t, St
) = Z
t
. For thenumerical approximation, we need to switch to log asset domain X
t
= logSt
, with
dXt
= (µ� 1
2�
2)dt+ �dWt
.
We set µ = µ� 12�
2, thusdX
t
= µdt+ �dWt
,
52
and Y
t
= v(t, eXt), Zt
= �e
X
t
v
s
(t, eXt).We take the following parameter values
X0 = log(100), K = 100, r = 0.1, µ = 0.2, � = 0.25, T = 0.1,
with the exact solutions Y0 = v(t0, S0) = 3.65997 and Z0 = �S0vs(t0, S0) = 14.14823.For the results of the BCOS method, we refer to the Figure 8.2 for the approximatedvalue
Figure 5.1.: In this example, we take N = 29, timesteps M = 10. On the left graph we see the
error of Y (t0, x0) and on the right, the error of Z(t0, x0)
53
A. Appendix
Theorem A.1. [14] Burkholder-Davis-Gundy inequalities
This gives bounds for the maximum of a martingale in terms of the quadratic varia-tion. For a local continuous martingale M
t
=Rt
0 �sdWs
starting at zero, with maximumdenoted by M
⇤t
= supst
|Ms
|, and any real number p � 1, the inequality is
c
p
E✓Z
t
0�
2s
ds
◆ p
2
E(M⇤t
)p C
p
E✓Z
t
0�
2s
ds
◆ p
2
.
Here, c
p
< C
p
are constants depending on the choice of p, but not depending on themartingale M or time t used and p > 0.
Theorem A.2. [10] Martingale’s Representation Theorem
For 0 t T , let W
t
be a Brownian motion on (⌦,F ,P) and Ft
be the filtrationgenerated by this Brownian motion. Let M
t
be a square-integrable martingale (under P)relative to this filtration. Then there exists a unique adapted process Z
t
such that
M
t
= M0 +
Zt
0Z
s
dWs
, 0 t T.
In particular, the paths of M are continuous.
Lemma A.3. [5] Gronwall’s LemmaLet T > 0, c � 0, u(·) a measurable non negative function in [0, T ], and v(·) anintegrable non negative function in [0, T ]. If
u(t) c+
Zt
0v(s)u(s)ds, 8t 2 [0, T ],
then
u(t) c exp
✓Zt
0v(s)ds
◆.
Theorem A.4. Young’s inequality
Let a, b be non negative real numbers and p, q 2 (0,1] such that 1p
+ 1q
= 1. Then
ab a
p
p
+b
q
q
.
For p = q = 2, we have
ab a
2
2+
b
2
2and for an ✏ > 0 we have the so-called Peter–Paul inequality.
ab a
2
2✏+✏b
2
2.
54
Lemma A.5. [3] For a, b 2 R it holds
|a+ b|2 2|a|2 + 2|b|2
Proof. This result is a direct conclusion from Young’s inequality (A.4):
|a+ b|2 |a|2 + 2|a||b|+ |b|2 2|a|2 + 2|b|2
Lemma A.6. [3] For positive, real numbers �, µ > 0 and C, y, z, t � 0, the inequality
2y(t+ Cz) Cz
2
�
2+ Cy
2�
2 +t
2
µ
2+ y
2µ
2
is valid.
Proof. By Young’s inequality (A.4), it follows that
2yt t
2
µ
2+ y
2µ
2
2yCz Cz
2
�
2+ Cy
2�
2,
thus
2y(t+ Cz) Cz
2
�
2+ Cy
2�
2 +t
2
µ
2+ y
2µ
2.
Theorem A.7. [5] Ito isometry
For a local continuous martingale M
t
=Rt
0 �sdWs
starting at zero. The stochastic inte-gral satisfies the Ito isometry
E✓Z
t
0�
s
dWs
◆2
= EZ
t
0�
2s
ds,
which holds when H is bounded or, more generally, when the integral on the right handside is finite.
Lemma A.8. [1] Doob’s inequalityLet S
t
= sup0st
X
s
and for p � 1 and
kXt
kp
= (E [|Xt
|p]) 1p
.
where X is a martingale, for p > 1,
kXT
kp
kST
kp
p
p� 1kX
T
kp
.
55
Theorem A.9. [16] Banach Fixed Point Theorem
Let H be a non-empty Banach space and f : H ! H be a contraction on H (i.e. there isa real number c such that c 2 (0, 1) and for any x, y 2 H; ||f(x)� f(y)|| c||x� y||)).Then f has only one fixed point.
Theorem A.10. [17] Let Z1, Z2, ..., Zn
be independent random variables with Z
i
⇠N (0, 1). If Y =
Pn
i=1 z2 then Y follows the chi-square distribution with n degrees of
freedom. We writeY ⇠ �
2n
Theorem A.11. [17] Let X1, X2, ..., Xn
be independent random variables with X
i
⇠N (µ,�). If Y =
Pn
i=1
�x
i
�µ
�
�2then Y follows the chi-square distribution with n degrees
of freedom. We writeY ⇠ �
2n
Theorem A.12. [14] Suppose that the coe�cients b(t, x),�(t, x) of the SDE dXt
=b(t,X
t
)dt+�(t,Xt
)dWt
are locally Lipschitz-continuous in the space variable; i.e., thereexists a constant C > 0 such that for every t � 0:
|b(t, x)� b(t, x0)|+ |�(t, x)� �(t, x0)| C|x� x
0|
|�(t, x)|+ |b(t, x)| C(1 + |x|)Then the SDE has indeed a strong solution.
56
Conclusion
In this thesis we considered a Backward SDEs (BSDEs), where the generator satisfies theLipschitz condition. In order to find a F
t
-adapted solution to the BSDE an additionalFt
-adapted process Z has to be introduced, which results in the fact that the solution of aBSDE consists of a pair of processes (Y, Z). By the existence and uniqueness theorem weproved that this pair of processes indeed exists and is unique. Solving BSDEs are used instochastic optimal control problems and hedging problems in finance where Y representsthe dynamics of the value of the replicating portfolio and Z corresponds to the hedgingstrategy. When the solution of a BSDE is associated with a state process satisfyinga forward SDE, we showed that this Forward-Backward SDE satisfies the Markovianproperty. By the famous Feynman-Kac formula Markovian BSDEs yield a representationformula for semi-linear parabolic PDEs. Conversely, under smoothness assumptions (i.e.viscosity solution), the solution of the BSDE corresponds to the solution of a semi-linearparabolic PDEs.A drawback of BSDEs is that they can rarely be solved explicitly, so simulation of
BSDEs is of big importance. We introduce a probabilistic numerical method for solvingbackward stochastic di↵erential equation, whereas the design of numerical schemes forBSDEs is an important task. Since BSDEs are terminal value problems for stochasticdi↵erential equations, a natural time discretization for BSDEs works backwards in time.However, the solution must be adapted to the information, which makes the construc-tion of numerical solutions to BSDEs a more challenging problem compared to initialvalue problems for stochastic di↵erential equations. In order to keep the time discretizedsolution adapted to the filtration, the time discretization scheme requires nestings of theconditional expectation. In the next step, an estimator for the conditional expectationhad to be applied. We studied the method, called BCOS method, developed in [26].This method is based on Fourier-cosine series expansions and relies on the character-istic function of the transitional probability density function. This approach allowedus to approximate the conditional expectations in an e�cient way. When the char-acteristic function of the underlying process cannot be derived easily, we can use thecharacteristic function of a discrete forward process to approximate the solution, wherewe approximated the underlying forward SDE by the Euler scheme. The applicabilityof the resulting method is therefore quite general. A numerical test demonstrates theapplicability of the BCOS method for BSDEs in economic and financial problems.
57
Popular summary
Whereas the theory and applications of classical forward stochastic di↵erential equations(FSDE), with a known initial value, is traditional, in this thesis we studied backwardstochastic di↵erential equations (BSDEs). A BSDE is a stochastic di↵erential equationfor which a terminal condition, instead of an initial condition, is known beforehand.In recent years, BSDEs have received more attention in mathematical finance and eco-
nomics, for example, the Black-Scholes formula for pricing options can be represented bya system of (decoupled) Forward-Backward SDE. The solution of a BSDE consists of apair of processes. However as in most of the cases it is not possible to analytically solveBSDEs, numerical methods are in great demand. These so-called probabilistic numeri-cal methods for solving BSDEs can be divided into two steps: a time discretization ofthe stochastic process and approximations for the appearing conditional expectations. Inthis thesis we employ a general theta-method for the time-integration and propose an ap-proach based on the Fourier series, more precise on Fourier-Cosine series, to approximatethe solution backwards in time. This method links the computation of the expectationsto the characteristic function of the transitional probability density function, which iseither given or easy to approximate. We call the method the BCOS method, short forBSDE-COS method.
58
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