Backward stochastic differential equations with applications

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Computational Engineering and Technical Physics Techno-Mathematics

Arnold Kaynet Muchatibaya

BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH APPLICATIONS

Master’s Thesis

Examiners: Professor Simo Särkkä Professor Heikki Haario Supervisors: Professor Simo Särkkä

Professor Heikki Haario

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Lappeenranta University of Technology School of Engineering Science

Computational Engineering and Technical Physics Techno-Mathematics

Arnold Kaynet Muchatibaya

Backward Stochastic Differential Equations with Applications

Master’s Thesis 2018

60pages.

Examiners: Professor Simo S¨arkk¨a Professor Heikki Haario

Keywords: Stochastic differential equations, Backward stochastic differential equations, Doob’s h-transform, Feynman-Kac, L´evy process

In this thesis we study backward stochastic differential equations driven by a Brownian motion and by a L´evy process and their applications, focusing on their applications to fi- nancial markets. We give results on the existence and uniqueness of solution of backward stochastic differential equations when the drift is Lipschitz continuous and the terminal condition is square integrable and measurable with respect to the terminal filtration.

Backward stochastic differential equations associated with a forward stochastic differential equation are investigated. We use the generalisation of the Feynman-Kac formula to show the relationship between a backward stochastic differential equation associated with a forward stochastic differential equation and a partial differential equation in the Brown- ian motion case and a partial differential integral equation for the L´evy process case. The Doob’s h-transform is studied for the Brownian motion and applied to stochastic differential equations. Finally, we conclude with an application to option pricing and hedging of a European calls for both Brownian and L´evy processes.

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All glory and praise be given to the most High for giving me the life and sustaining me during my studies until the completion of this thesis. I wish to thank my supervisors Professor Simo S¨arkk¨a and Professor Heikki Haario for their help, resources and super- vision throughout the preparation of this thesis. I would also like to thank my family for the strength and support during my studies. Finally, thank you to African Institute for Mathematical Sciences (AIMS-Tanzania) and Lappeenranta University of Technology for arranging and awarding of the scholarship to undertake studies at the later.

Lappeenranta, October 19, 2018

Arnold Kaynet Muchatibaya

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ABBREVIATIONS AND SYMBOLS

These are the abbreviations we will use throughout this essay:

a.s. Almost surely C Complex numbers A∈B Element A in set B

∀ For all

⇒ Implies

h·,·i Inner product 7→ Maps to

|| · || Norm of (·) R Real numbers

∃ There exists

∪ Union of sets w.r.t with respect to

BSDE Backward stochastic differential equation EMM Equivalent martingale measure

FPK Fokker-Planck-Kolmogorov

FBSDE Forward-backward stochastic differential equation FSDE Forward stochastic differential equation

ODE Ordinary differential equation PDE Partial differential equation

PDIE Partial differential integral equation SDE Stochastic differential equation

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1 INTRODUCTION

Allowing randomness in the coefficients of an ordinary differential equation results in realistic mathematical models of physical phenomena. Stochastic differential equations arise when we allow randomness in the coefficients of ordinary differential equations or when the forcing is an irregular stochastic process like Gaussian white noise [1]. There are two types of stochastic differential equations; if the initial condition is specified, we have a forward stochastic differential equation and if the terminal condition is specified, we have a backward stochastic differential equation.

The theory of backward stochastic differential equations has found wide applications in areas such as stochastic optimal control, theoretical economies, and mathematical finance problems such as the theory of hedging and non-linear pricing in incomplete markets [2].

Backward stochastic differential equations can be driven by a L´evy process, Brownian motion, Poisson process, or a combination of these.

Bismut [3] first introduced backward stochastic differential equations in a linear form as the equation for the conjugate variable in the stochastic version of the Pontryagin maximum principle. Pardoux et al. [4] were the first to consider general backward stochastic differential equations. Their main result was the existence and uniqueness of an adapted pair of processes as a solution of a backward stochastic differential equation.

Several authors have extended their results. Peng [5] used backward stochastic differential equations to obtain a probabilistic interpretation for systems of second order quasi- linear parabolic partial differential equations. Pardoux et al. [6] introduced a new class of backward stochastic differential equations, which allowed them to produce a probabilistic representation of a certain quasi-linear stochastic partial differential equation thus ex- tending the Feynman-Kac formula for stochastic partial differential equations. Antonelli [7] showed the existence and uniqueness of a solution of a backward stochastic differential equation inspired from the stochastic differential utility in finance theory. Ma et al.

[8] investigated adapted solutions to a class of forward-backward stochastic differential equations in which the forward stochastic differential equation is non-degenerate. They showed that the adapted solution can be sought over an arbitrarily prescribed time du- ration via a direct four step scheme. Using this scheme, they proved that the backward components of the adapted solution are determined explicitly by the forward component via the solution of a certain quasi-linear parabolic partial differential equation system.

El-Karoui et al. [9] summarized the existence and uniqueness of solutions of backward

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stochastic differential equations by Pardoux et al. [4] and gave new shorter proofs. They stated the a priori estimates of the difference of two backward stochastic differential equations, and the uniqueness and existence was proved using a fixed point theorem. They also looked at the solution of a backward stochastic differential equation associated with a forward stochastic differential equations. The main property was that the solution is Marko- vian in the sense that it can be written as a function of time and state process. The generalisation of the Feynman-Kac formula is given, and they also showed that under smoothness assumptions, the solution of the backward stochastic differential equation corresponds to a solution of a system of quasi-linear parabolic partial differential equations. These results could be applied to option pricing of a European call in the constrained Markovian cases.

Buckdahn and Pardoux [10] proved the existence and uniqueness of a solution to a backward stochastic differential equation with respect to both the Brownian motion and Pois- son random measure and the associated integro-partial differential equation of parabolic type. They proved that under certain conditions, the solution of a backward stochastic differential equation provides the unique viscosity solution of the associated integro-partial differential equation. Situ [11] studied backward stochastic differential equations driven by Brownian motion and Poisson point process. A new existence and uniqueness result for the solution of the partial differential integral equation with non Lipschitz force is obtained. Oukine [12] considered a backward stochastic differential equation driven by a Poisson random measure. The integral representation of the square integral random variable in terms of a Poisson random measure is the main result.

Nualart and Schoutens [13] proved a martingale representation theorem for a L´evy process satisfying some exponential moment condition. Nuarlart et al. [14] used results from [13] to establish existence and uniqueness of solutions for backward stochastic differential equations driven by a L´evy process. Our work is primarily based on [9] and [14].

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1.1 Objectives

This thesis is based on the articles by Nualart et al. [13] and El-Karoui et al.[9]. We expand the proof studied in the articles, especially the existence and uniqueness of solutions of a BSDE under Lipschitz conditions on the drift driven by a L´evy process and Brownian motion. We are also concerned with the application of these BSDEs in finance.

To achieve our purpose, we consider some specific objectives:

(i) We study the existence and uniqueness of a general BSDE under the Lipschitz condition on the drift driven by a L´evy process and Brownian motion expanding the proofs in these sections.

(ii) Apply the Feynman-Kac formula to the BSDEs to get the relationship between BSDEs, PDEs and PDIEs.

(iii) Consider the application of the theory above to European call options.

(iv) We study the Doob’s h-transform applied to an SDE in the Brownian Motion case.

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1.2 Outline

In Chapter2, we study the existence and uniqueness of a solution of BSDEs with a Lips- chitz driver and driven by a L´evy process and Brownian motion in Section2.3. We also consider the Feynman-Kac formula for the BSDEs to establish the relationship with the partial differential equation in the case of BSDE driven by Brownian motion and the partial differential integro equation in the case of the BSDE driven by a L´evy process in Section2.4. In Section2.5we study the Doob’s h-transform and application to stochastic differential equations to come up with BSDE. In Chapter 3, we look at the application of these studied BSDEs to option pricing. In Chapter 4, we discuss the results we have obtained in our work and mention possible future work to be done. Finally we conclude our work in Chapter5.

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2 BACKWARD STOCHASTIC DIFFERENTIAL EQUA- TIONS

2.1 Preliminaries

In this section we will give the background of SDEs, theorems and inequalities which will be necessary to refer in the forthcoming chapters. We start by defining the space we will be working on,

Definition 1(Probability space). IfΩis a given set, then aσ-algebraF onΩis a family F of subsets ofΩwith the following properties:

(i) ∅ ∈ F,

(ii) F ∈ F ⇒F^c ∈ F whereF^cis the complement ofF inΩandF^c = Ω\F,and (iii) A1, A2,· · · ∈ F ⇒ A:=S∞

i=1∈ F.

Then the pair(Ω,F)is called a measurable space. A probability measurePon(Ω,F)is a functionP:F 7−→[0,1]such that

(i) P(∅) = 0, P(Ω) = 1,and

(ii) ifA₁, A₂,· · · ∈ F and{Ai}^∞_i=1is disjoint, then

P

∞

[

i=1

!

=

∞

X

i=1

P(Ai).

Then the triplet(Ω,F,P)is a probability space [15].

A martingale is a stochastic process for which at a time in the realised sequence the expectation of the next value is the current observed value given prior observation. Now we give the mathematical definition is as follows.

Definition 2(Martingale). A filtration on(Ω,F)is a familyF = {Ft}_t≥0 ofσ-algebras Ft⊂ F such that

0≤s≤T ⇒ Fs ⊂ Ft.

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An n-dimensional stochastic process {Xt}_t≥0 on (Ω,F,P) is called a martingale [15]

with respect to a filtration{Ft}_t≥0(and with respect toP) if

(i) XtisFt−measurable for allt, (ii) E[|Xt|]<∞for allt,and (iii) E[Xs|Ft] =Xta.s. for alls ≥t.

Now let us define a L´evy process.

Definition 3 (L´evy process). Given the probability space in Definition 1, a L´evy pro- cessX = {Xt, t ≥ 0}taking values inR^d is a stochastic process having stationary and independent increments and we always assumeX0 = 0with probability1.So

• Xt: Ω−→R^d.

• Given any selection of distinct time points 0 ≤ t₁ < t₂ < · · · < tn the random vectorsXt1, Xt2 −Xt1, Xt3 −Xt2,· · · , Xtn−Xtn−1 are all independent.

• Given any two distinct times0≤s < t <∞,the probability distribution ofXt−Xs

coincides with that ofXt−s.[16]

Brownian motion and Poison process are examples of L´evy processes. Thus we have,

Definition 4 (Brownian motion). A standard Brownian motion in R^d is a L´evy process W = (Wt, t ≥0)for which

• Wt∼N(0, tI)for eacht≥0

• W has continuous sample paths

Every L´evy process is characterised by its characteristic function which is defined as follows [7].

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Definition 5(Characteristic function). LetXbe a random variable defined on the probability space in Definition1taking values inR^dwith the probability lawP_x.It’s characteristic functionφx :R^d−→Cis

φx(u) =E e^i(u,x)

= Z

Ω

e^i(u,X(ω))P(dω)

= Z

R^d

e^i(u,y)P_x(dy)

for eachu∈R^d

We define the indicator functionχA as a function defined to be1onAand0elsewhere.

Now the characteristic function is given by the L´evy -Khintchine formula defined below as.

Definition 6(L´evy-Khintchine formula). IfX = {Xt, t ≥ 0}is a L´evy process, then it has a specific form for its characteristic function [16]. More precisely∀t≥0, u∈R^d

E(e^i(u,X^t⁾) =e^tη(u) where

η(u) =i(b, u)− 1

2(u, au) + Z

R^d−{0}

e^i(u,y)−1−i(u, y)χ0<|y|<1(y)

ν(dy).

where

• b∈R^d

• ais a positive definite symmetricd×dmatrix

• νis a L´evy measure onR^d− {0}so that Z

R^d−{0}

min{1,|y|²ν(dy)}<∞.

A Lévy process can be decomposed into a linear drift, Brownian motion, and a pure jump process [16]. This result is called the Lévy-Itô decomposition and defined as follows.

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Theorem 2.1.1(The Lévy-Itô decomposition). IfX is a Lévy process, then there exists b ∈ R^d,a Brownian motionB with diffusion matrixQand an independent Poisson ran- dom measureN onR⁺×(R^d− {0})such that for eacht ≥0,

Xt =bt+Bt+ Z

|x|<1

xN˜(t, dx) + Z

|x|>1

xN(t, dx).

Proof. See [16]

Definition 7. Let us consider the SDE,

dXt=b(t, x)dt+σ(t, x)dWt

X(0) =x, t≥0.

A strong solution of this SDE on the given probability space with respect to the fixed Brownian motion W and initial condition x is a process X = {Xs; 0 ≤ s ≤ T} with continuous sample paths and with the following properties:

(i) Xis adapted to the filtrationFs, (ii) P[X₀ =x] = 1,

(iii)

P





T

Z

0

{|b(s, Xs)|+σ²_ij(s, Xs)}ds <∞



= 1

holds for every1≤i≤d,and (iv) The integral version is

Xs =X0+

T

Z

0

b(s, Xs)ds+

T

Z

0

σ(s, Xs)dWs.

[17].

For any martingale adapted with respect to a Brownian motion can be expressed as an Itˆo integral with respect to the same Brownian motion as follows.

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Theorem 2.1.2(Martingale representation). Let(Wt,0≤ t≤ T)be a Brownian motion on(Ω,F,P). Let{Ft; 0 ≤ t ≤ T}be the filtration generated by this Brownian motion.

Let{Xt; 0≤t ≤T}be a martingale (underP) relative to this filtration (i.e., for everyt, Xtis Ft−measurable, and for 0 ≤ s ≤ t ≤ T,E[Xt|Fs] = Xs a.s.). Then there is an adapted process{At; 0≤t≤T},Atsquare integrable such that

Xt =X₀+

t

Z

0

AudWu, 0≤t ≤T.

Proof. See [15]

The Burkholder-Davis-Gundy inequalities relate the maximum of a local martingale with it’s quadratic variation. This result is important in the proofs in the next chapters.

Theorem 2.1.3 (Burkholder-Davis-Gundy inequalities). Let T > 0 and (Mt)0≤t≤T be a continuous local martingale such that M0 = 0. For every 0 < p < ∞, there exists universal constantscp, Cp independent ofT and(Mt)0≤t≤T such that,

cpE hMti

p 2

T

≤E

sup

0≤t≤T

|Mt| p

≤CpE hMti

p 2

T

.

Proof. See [17].

The Banach fixed point theorem is important in the proof of existence and uniqueness of solution of a BSDE, hence we first define a metric space then give theorem.

Definition 8. Let (X, d) be a metric space. A mapping T : X 7−→ X is Lipschitz continuous if there exists a constant α > 0 such that d(T x, T y) ≤ α d(x, y) for all x, y ∈X.If0≤α <1, thenT is called a contraction mapping, andαis called the factor ofT [18].

Theorem 2.1.4(Banach fixed point theorem). Suppose that(X, d)is a generalised com- plete metric space, and that the functionT :X 7−→X is a contraction.

Letx0 ∈X,and consider the sequence of successive approximations with initial element x0

x0, T x0, T²x0,· · · , Tⁱx0,· · ·. (1) Then either

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1. For every integeri= 0,1,2,· · ·,one has

d(Tⁱx0, Tⁱ⁺¹x0) =∞, or

2. The sequence of approximations, Equation(1)isd-convergent to a fixed point ofT.

Proof. See [18]

We need to define anL^p space before the H¨older inequality as follows.

Definition 9. Consider the measurable space in Definition 1 and 1 ≤ p, q ≤ ∞. The spaceL^p(Ω)consists of equivalence classes of measurable functionsf : Ω7→Rsuch that

Z

|f|^pP(dω ∈ F)<∞,

where ω ∈ Ω and two measurable functions are equivalent if they are equalP a.e [19].

TheL^p norm off ∈L(Ω)is defined by

||f||L^p = Z

|f|^pP(dω∈ F) ¹_p

. Whenp= 1the spaceL¹consists of all integrable functions onΩ.

Theorem 2.1.5(H¨older inequality). Assume that a functionf ∈ L^p andg ∈ L^q,where p, q∈(1,∞)are conjugate numbers, that is,

1 p+ 1

q = 1.

Thenf g ∈L¹,and the following inequality holds

Z

f g dx

≤ Z

|f g|dx≤ ||f||p||g||q. (2)

Proof. See [19]

Theorem 2.1.6(Markov inequality). Suppose(Ω,F,P)is a measure space,f is a mea- surable extended real valued function, andε >0.Then

P({ω∈Ω :|f(ω)| ≥ ε})≤ 1 ε

Z

Ω

|f|P(dω ∈ F).

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Proof. See [19]

Definition 10(Itˆo process). LetWtbe a one-dimensional Brownian motion on(Ω,F,P).

An Itˆo process (or stochastic integral) is a stochastic processXton(Ω,F,P)of the form

Xt=X0+

t

Z

0

b(s, ω)ds+

t

Z

0

σ(s, ω)dWs,

where

P





t

Z

0

σ(s, ω)²ds <∞for allt≥0



= 1,

and

P





t

Z

0

|b(s, ω)|ds <∞for allt≥0



= 1.

[15]

We define the quadratic variation and cross variance as follows.

Definition 11. IfXt(·) : Ω7−→ Ris a continuous stochastic process, then forp > 0the p’th variation process ofXt;hX, Xi^(p)_t is defined by

hX, Xi^(p)_t (ω) = lim

△tk→0

X

tk≤t

|Xtk+1(ω)−Xtk(ω)|^p(limit in probability)

where0 = t1 < t2 < · · ·< tn = tand△tk =tk+1−tk. Ifp = 1then it is called total variation and ifp= 2is called quadratic variation. We also have

hX, Xi^(p)_t =hX, Xit=hXit.

More generally for the cross variation between two processesXtandYtwe have hX, Yit(ω) = lim

△tk→0

X

t_k≤t

(Xt_k+1(ω)−Xt_k(ω))(Yt_k+1(ω)−Yt_k(ω)).

[15].

Let us consider the Itô formula for Brownian motion and Lévy process driven Itô process as follows.

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Theorem 2.1.7 (Itˆo formula). Let Xt be an Itˆo process, and let f(t, x) be a function for which the partial derivativesft, fx, fxx are defined and continuous. Then for every T ≥0,

f(T, Xt) =f(0, X0) +

T

Z

0

ftdt+

T

Z

0

fxdXt+ 1 2

T

Z

0

fxxdhXit. (3) Letf(t, x, y)be a function whose partial derivativesft, fx, fy, fxx, fxy, fyy are defined and are continuous. LetXtandYtbe Itˆo processes. The two dimensional Itˆo formula in differential form is

df(t, x, y) =ftdt+fxdXt+fydYt+1

2fxxdhXit+fxydhX, Yit+ 1

2fyydhYit. (4) Proof. See [20]

Theorem 2.1.8 (Itˆo formula for L´evy process driven SDEs). Let X = (X¹,· · · , Xⁿ) be an n-tuple of semi martingales and let f : Rⁿ 7−→ R have continuous second order partial derivatives. Thenf(x)is a semi-martingale and the following formula holds

f(Xt)−f(X₀) =

n

X

i=1 t

Z

0⁺

∂f

∂xi

(X_s−)dX_s⁽ⁱ⁾+1 2

X

1≤i,j≤n t

Z

0⁺

∂²f

∂xi∂xj

(X_s−)d[X⁽ⁱ⁾, X^(j)]^c_s

+ X

0<s≤t

(

f(Xs)−f(Xs−)−

n

X

i=1

∂f

∂xi

(Xs−)△X_s⁽ⁱ⁾ )

Proof. See [21]

Theorem 2.1.9 (Doob’s maximal inequality). Let (Xt; Ft, 0 ≤ t < ∞) be a semi- martingale whose every path is right-continuous. Letα < β be real numbers, and[0, T] is a sub interval of[0,∞).Then

E

sup

0≤t≤T

Xt

p

≤ p

p−1 p

E(X_T^p), p >1, providedXt≥0a.s. Pfor everyt≥0andE(X_T^p)<∞.

Proof. See [17]

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Theorem 2.1.10(Gronwall’s inequality). Letg(t)andh(t)be regular non-negative func- tions on[0, T].Then for any regularf(t)≥0satisfying the inequality for all0≤t≤T,

f(t)≤g(t) +

t

Z

0

h(s)f(s)ds,

we have

f(t)≤g(t) +

t

Z

0

h(s)g(s) exp





t

Z

s

h(u)du



ds. (5) In particular, if g is non-decreasing, Equation(5)simplifies to give

f(t)≤g(t)e^R⁰^t^h(s)^ds. In its simplest form wheng =Aandh=B are constants,

f(t)≤Ae^Bt.

Proof. See [22]

Theorem 2.1.11 (Fubini’s theorem). Let f(x, t) be continuous on [t, T]×[0, b]. Then iterated integrals:

T

Z

t b

Z

0

f(x, t)dx dt=

b

Z

0 T

Z

t

f(x, t)dt dx.

Proof. See [23]

We consider the Fubini theorem for stochastic processes. We first of all define an FV process and some notation we will use for the theorem.

Definition 12. An FV process is a cadlag adapted stochastic process such that all its paths are of finite variation on each compact interval onR₊[21].

The following Notation is by [21]. Notation: Let A be an FV process and letF be jointly measurable process such that

t

Z

0

F(s, ω)dAs(ω)

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exists and is finite∀t >0,a.s. we let

(F ·A)t(ω) =

t

Z

0

F(s, ω)dAs(ω).

We also writeF ·Ato denote the processF ·A= (F ·At)t≥0. Then the Fubini theorem is as follows.

Theorem 2.1.12(Fubini’s theorem (stochastic processes)). LetX be a semi-martingale, H_t^a =H(a, t, ω)be a boundedA ⊗Pmeasurable function, and letµbe a finite measure on A. Let Z_t^a = Rt

0 H_sâdXs be A ⊗ B(R₊)⊗ F measurable such that for each a, Zâ is cádlág version of Hâ·X.Then Yt = R

AZ_tâµ(da)is cádlág version of H ·X, where Ht=R

Aµ(da)[21].

The Itˆo isometry is essential for computation of variances of random variables given as an Itˆo integral. The following theorem is by [20].

Theorem 2.1.13 (Itˆo Isometry). Let W : [0, T]×Ω −→ R be the standard Brownian motion defined to T > 0. Let X : [0, T]× Ω −→ R be a stochastic process that is adapted to the natural filtration of the Brownian motion then,

E











T

Z

0

XsdWs





2



=E





t

Z

0

X_s²ds





Theorem 2.1.14 (Comparison theorem). Let (f¹, ξ¹) and (f², ξ²) be two standard pa- rameters of BSDEs, and let(X¹, Y¹) and(X², Y²)be the associated square integrable solutions. We suppose that

1. ξ¹ ≥ξ² Pa.s.

2. δ2ft =f¹(t, X_t², Y_t²)−f²(t, X_t², Y_t²)≥0, dPN dt a.s.

Then we have almost surely for any timet,

X_t¹ ≥X_t².

Moreover, the comparison is strict. That is, if, in addition, X₀¹ = X₀², then ξ¹ = ξ², f¹(t, X_t², Y_t²) = f²(t, X_t², Y_t²), dPN

dt a.s., and X¹ = X² a.s. More generally,

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ifX_t¹ =X_t²on a setA∈ Ft, thenX_s¹ =X_s²almost surely on[0, T]×A, ξ¹ =ξ²a.s. on A, andf¹(t, X_t², Y_t²) = f²(t, X_t², Y_t²)onA×[t, T]dPN

dta.s.

Proof. See [9]

Definition 13 (Generalised generator). The generalised (infinitesimal) generator for a time dependent functionφ(x, t)can be defined as [24]

Atφ(x, t) = lim

s↓0

E[φ(x_t+s, t+s)]−φ(xt, t)

s (6)

for a time dependent SDE and At(·) = ∂(·)

∂t +X

i

∂(·)

∂xi

bi(t, Xt) + 1 2

X

i,j

∂²(·)

∂xi∂xj

[σ(t, Xt)Qσ^∗(t, Xt)]ij

whereσ^∗ is the transpose ofσand the SDE is

dXt=b(t, Xt)dt+σ(t, Xt)dWt, whereQis the diffusion matrix of Brownian motion.

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2.2 Background

In this section we will discuss the theory of BSDEs. First, we give some notations used in this chapter and the rest of the work, by [9]. Forx∈R^d,|y|denotes the Euclidean norm, andhy, zidenotes the inner product. Ann×dmatrix will be considered as an element y∈R^n×d,and the Euclidean norm is given by

|z| =p

trace(zz^∗), wherez^∗ is the transpose ofz, and

hy, zi=trace(yz^∗), fory, z∈R^n×d.

Given a probability space, andRⁿ-valued Brownian motionW, we consider the following definitions:

• {Ft; 0 ≤ t ≤ T},the filtration generated by the Brownian motionW and, P the σ-field of predictable sets ofΩ×[0, T].

• L²_T(R^d),the space of allFT measurable random variablesX : Ω7−→R^dsatisfying

||Y||² =E(|Y|²)<+∞.

Usually denoted asL^2,d

T .

• H²

T(R^d),the space of all predictable processesϕ : Ω×[0, T]7−→R^dsuch that

||ϕ||² =E

T

Z

0

|ϕt|²dt <+∞.

Usually denoted asH^2,d

T .

• H¹_T(R^d),the space of all predictable processesϕ : Ω×[0, T]7−→R^dsuch that

E v u u u t

T

Z

0

|ϕt|²dt < +∞.

Usually denotedH^1,d

T .

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• Forβ > 0andφ ∈H²

T(R^d),

||φ||²_β =E

T

Z

0

e^βt|φt|²dt,

andH²

T,β(R^d)denotes the spaceH²

T(R^d)equipped with the norm || • ||β. Usually denotedH^2,d

T,β.

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2.3 Existence and Uniqueness

In this section we are going to prove the existence and uniqueness of the solution of a BSDE driven by L´evy process and another driven by Brownian motion. The drift for both BSDEs is considered to be Lipschitz.

2.3.1 Brownian Motion

Consider a BSDE of the form [9],

−dYt=f(t, Yt, Zt)dt−Z_t^∗dWt, YT =ξ. (7) or equivalently,

Yt=ξ+

T

Z

t

f(s, Ys, Zs)ds−

T

Z

t

Z_s^∗dWs (8)

where

• The terminal value is anFT-measurable random variable,ξ: Ω7−→R^d.

• The generatorf : Ω×R⁺×R^d×R^n×d7−→R^dand isPN B^dN

B^n×d-measurable.

HereB^ddenotes Borel-measurable sets inR^d,andB^n×ddenotes Borel-measurable sets in R^n×d.

Definition 14. A solution of Equation (7) is a pair (Y, Z) such that {Yt ; t ∈ [0, T]}

is a continuous R^d-valued adapted process, and {Zt ; t ∈ [0, T]} is an R^n×d-valued predictable process satisfying

T

Z

0

|Zs|²ds <+∞, Pa.s.

Definition 15. Suppose thatξ ∈L^2,d

T ,f(·,0,0)∈H^2,d

T , andf is uniformly Lipschitz that is, there existsC > 0such thatdPN

dta.s.

|f(ω, t, x1, y1)−f(ω, t, x2, y2)| ≤C(|x1−x2|+|y1−y2|), ∀(x1, y1),∀(x2, y2)∈R². Then(f, ξ)are said to be standard parameters for the BSDE.[9]

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Proposition 1. Let ((fⁱ, ξⁱ);i = 1,2) be two standard parameters of the BSDE and ((Yⁱ, Zⁱ);i= 1,2)be two square integrable solutions. LetCbe a Lipschitz constant for f¹,and putδYt = Y_t¹ −Y_t² andδ2ft =f¹(t, Y_t², Z_t²)−f²(t, Y_t², Z_t²).For any(λ, µ, β) such thatµ >0, λ² > C, β ≥C(2 +λ²) +µ²,it follows that

||δY||²_β ≤T

e^βTE(|δYT|²) + 1

µ²||δ₂f||²_β

, (9)

||δZ||²_β = λ² λ²−C

e^βTE(|δYT|²) + 1

µ²||δ₂f||²_β

. (10)

[9].

The Proof of Proposition1is done by [9] and is a follows.

Proof. Let(Y, Z) ∈H^2,d

T ×H²_T(R^n×d)be a solution of Equation (7). From Equation (8) using triangle inequality and H¨older inequality Theorem2.1.5, we have

|Yt| ≤ |ξ|+

T

Z

t

f(s, Ys, Zs)ds

+

T

Z

t

Z_s^∗dWs

≤ |ξ|+

T

Z

t

|f(s, Y s, Zs)|ds+

T

Z

t

Z_s^∗dWs

.

Now taking the supremum, we have

sup

0≤t≤T

|Yt| ≤ |ξ|+

T

Z

0

|f(s, Ys, Zs)|ds+ sup

0≤t≤T

T

Z

t

Z_s^∗dWs

. (11)

We claim that Yt ∈ L^2,d

T .In fact, it is enough to show that each component of the right hand side of Equation (11) is in L^2,d

T . Thus using the triangular inequality, Itˆo Isometry

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(Theorem2.1.13)and the Burkholder-Davis Gundy inequality (Theorem2.1.3), we get

E





 sup

0≤t≤T

T

Z

t

Z_s^∗dWs

2



=E





 sup

0≤t≤T

T

Z

0

Z_s^∗dWs−

t

Z

0

Z_s^∗dWs

2





≤E





 sup

0≤t≤T

2







T

Z

0

Z_s^∗dWs

2

+

t

Z

0

Z_s^∗dWs

2











= 2E







T

Z

0

Z_s^∗dWs

2



+ 2E



 sup

0≤t≤T

t

Z

0

Z_s^∗dWs

2



= 2E





T

Z

0

|Zs|²ds



+ 2E



 sup

0≤t≤T

t

Z

0

Z_s^∗dWs

2



≤2E





T

Z

0

|Zs|²ds



+ 2C2E





T

Z

0

|Zs|²ds



.

Thussup_0≤t≤T |RT

t Z_s^∗dWs| ∈L^2,1

T .Sinceξ ∈L^2,1

T , fis uniformly Lipschitz,f(·,0,0)∈ H^2,1

T , X ∈H^2,d

T , Y ∈H^2,n×d

T ,then|ξ|+RT

0 |f(s, Xs, Ys)|ds ∈L^2,1

T ,thusYt∈L^2,1

T .

Now consider two solutions(Y¹, Z¹)and(Y², Z²)associated with(f¹, ξ¹)and(f², ξ²), respectively. LetδYs=Y_s¹−Y_s² such that

δYs=ξ¹−ξ²+

T

Z

t

[f¹(s, Y_s¹, Z_s¹)−f²(s, Y_s², Z_s²)]ds−

T

Z

t

δZ_s^∗dWs.

From the Itˆo’s formula Equation (3) applied froms =ttos =T to the semi-martingale e^βt|δYt|², we letf(t, x) =e^βt|x|² then substitutingxwithδYtandywithδZt,we have

e^βT|δYT|²−e^βt|δYt|² =

T

Z

t

βe^βs|δYs|²ds+

T

Z

t

2e^βshδYs, dδYsi+ 1 2

T

Z

t

2e^βsdhδYs, δYsi

=

T

Z

t

βe^βs|δYs|²ds+

T

Z

t

2e^βshδYs, dδYsi+

T

Z

t

e^βs|δZs|²ds.

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Then

e^βT|δYT|²−2

T

Z

t

e^βshδYs, dδYsi=e^βt|δYt|²+

T

Z

t

βe^βs|δYs|²ds

+

T

Z

t

e^βs|δZs|²ds.

But we have

2

T

Z

t

e^βshδYs, dδYsi=−2

T

Z

t

e^βshδYs,(f¹(s, Y_s¹, Z_s¹)−f²(s, Y_s², Z_s²))dsi

+ 2

T

Z

t

e^βshδYs, δZ_s^∗dWsi.

Thus we have

e^βt|δYt|²+

T

Z

t

βe^βs|δYs|²ds+

T

Z

t

e^βs|δZs|²ds

=e^βT|δYT|²+ 2

T

Z

t

−2

T

Z

t

e^βshδYs, δZ_s^∗dWsi.

Using the Burkholder-Davis Gundy inequality Theorem2.1.3, we have

E



 sup

0≤t≤T

T

Z

t

e^βsδZsδYsdW s



≤CE











T

Z

0

|δYs|²|δYs|²ds





1 2





≤C

E

sup

0≤t≤T

|δYs|²



E





T

Z

0

|δZs|²ds









1 2

<∞.

Sincesup_s≤T|δYs|belongs toL^2,1

T ,ande^βsδZsδYsbelongs toH^2,n

T ,and the stochastic integral

RT

t e^βshδYs, δZ_s^∗dWsiisP−integrable, with zero expectation. Using the triangle inequal-

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ity and Lipschitz condition onf,we have

|f¹(s, Y_s¹, Z_s¹)−f²(s, Y_s², Z_s²)|=|f¹(s, Y_s¹, Z_s¹)−f¹(s, Y_s², Z_s²) +f¹(s, Y_s², Z_s²)−f²(s, Y_s², Z_s²)|,

≤ |f¹(s, Y_s¹, Z_s¹)−f¹(s, Y_s², Z_s²)|

+|f¹(s, Y_s², Z_s²)−f²(s, Y_s², Z_s²)|,

≤C(|δYs|+|δZs|) +|δ2fs|.

Using2ab≤a²ǫ+ ^b_ǫ² applied to2yCz and2yt,we have the inequality, 2yCz ≤y²ǫ+ (Cz)²

ǫ . Takingǫ=Cλ² >0, we have

2yCz ≤y²Cλ²+ Cz² λ² . For2yt, we have

2yt≤y²ǫ+ t² ǫ. Takingǫ=µ² >0, we have

2yt≤y²µ²+ t² µ².

Now since2y(Cz+t) = 2Cyz+ 2yt, we have the inequality 2y(Cz+t)≤ Cz²

λ² + t²

µ² +y²(µ²+Cλ²). (12)

E

e^βt|δYt|² +βE





T

Z

t

e^βs|δYs|²ds



+E





T

Z

t

e^βs|δZs|²ds





=E

e^βT|δYT|² + 2E





T

Z

t





≤E

e^βT|δYT|² +E





T

Z

t

e^βs2hδYs,[C(|δYs|+|δZs|) +|δ₂fs|]ids





=E

e^βT|δYT|² +E





T

Z

t

e^βs(2ChδYs,|δYs|i+ 2ChδYs,|δZs|i+ 2hδYs,|δ₂fs|i) ds



.

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Using|hy, zi|=|y||z|, we have

E

e^βt|δYt|² +βE





T

Z

t

e^βs|δYs|²ds



+E





T

Z

t

e^βs|δZs|²ds





≤E

e^βT|δYT|² +E





T

Z

t

e^βs 2C|δYs|²+ 2|δYs|(C|δZs|+|δ₂fs|) ds



. Applying Equation (12) withy =|δYs|,Z =|δZs|,t =|δ₂fs|andC =C,we get

E

e^βt|δYt|² +βE





T

Z

t

e^βs|δYs|²ds



+E





T

Z

t

e^βs|δZs|²ds





≤E

e^βT|δYT|² +E





T

Z

t

e^βs

2C|δYs|²+ C|δZs|²

λ² + |δ₂fs|²

µ² +|δYs|² µ²+Cλ²

ds





=E

e^βT|δYT|² +E





T

Z

t

e^βs

C|δYs|²(2 +λ²) + C|δZs|²

λ² +|δ₂fs|²

µ² +µ²|δYs|²

ds





=E

e^βT|δYT|²

+ [C(2 +λ²) +µ²]E





T

Z

t

e^βs|δYs|²ds





+ C λ²E

T

Z

t

e^βs|δZs|²ds+ 1 µ²E

T

Z

t

e^βs|δ₂fs|²ds,

which gives

E[e^βt|δYt|²] + β−[C(2 +λ²) +µ²] E

T

Z

t

e^βs|δYs|²ds+

1− C λ²

E

T

Z

t

e^βs|δZs|²ds

≤E

e^βT|δYT|² + 1

µ²E

T

Z

t

e^βs|δ2fs|²ds. (13)

If we takeβ ≥[C(2 +λ²) +µ²]andC ≤λ²,we have

E[e^βt|δYt|²]≤E

e^βT|δYT|² +E

T

Z

t

e^βs|δ₂fs|² 1 µ²ds.

Taking the integral from0 −→ T, using Fubini’s theorem (Theorem2.1.11) and Defini-

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tion2.2, we have the control of the norm for the process|δY|as

||δY||²_β ≤T e^βTE[|δYT|²] +

T

Z

t

1

µ²||δ₂fs||²_βds

≤T

e^βTE[|δYT|²] + 1

µ²||δ₂fs||²_β

. The control of the process for|δZ|from Equations (13) is

λ²−C λ² E

T

Z

t

e^βs|δZs|²ds≤e^βTE[|δYT|²] + 1

µ²||δ2fs||²_β. That is,

λ²−C

λ² ||δZ||²_β ≤e^βTE[|δYT|²] + 1

µ²||δ₂fs||²_β. Hence,

||δZ||²_β ≤ λ² λ²−C

e^βTE[|δYT|²] + 1

µ²||δ₂fs||²_β

.

We have developed the necessary tools we need to prove uniqueness and existence of a solution. A detailed proof of the following theorem by [4], is given in their article. As in [9] we will prove it using the Banach fixed point theorem and a priori estimates.

Theorem 2.3.2. Given standard parameters (f, ξ), there exists a unique pair(Y, Z) ∈ H^2,d

T ×H²

T(R^n×d)which solves Equation(7)[9].

The solution is often referred to as a square-integrable solution.

Proof. This proof is by [9], we expand on the proof. We use the Banach fixed point theorem (Theorem2.1.4) for the mapping fromH^2,d

T,β×H²

T,βR^n×donto itself, which maps (y, z)onto the solution(Y, Z)of the BSDE with generatorf(t, xyt, zt),that is

Yt=ξ+

T

Z

t

f(s, ys, zs)ds−

T

Z

t

Z_s^∗dWs.

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The assumption that (f, ξ) are standard parameters implies f is uniformly Lipschitz, f(·,0,0) ∈ H^2,d

T , andξ ∈ L^2,d.Thus (f(t, yt, zt); t ∈ [0, T])belongs to H^2,d

T . Now we show why the solution to the BSDE is defined as a pair of adaptable processes. Consider the continuous versionM of a square integrable martingaleEh

ξ+RT

0 f(s, Ys, Zs)ds|Ft

i,

Mt =E



ξ+

T

Z

0

f(s, ys, zs)ds

Ft



.

[9]. By the Martingale representation theorem (Theorem 2.1.2) there exists a unique integrable processZ ∈H^2,n×d

T,β such that

Mt =M0+

t

Z

0

Z_s^∗dWs.

Define the adapted and continuous process [9]

Yt =Mt−

t

Z

0

f(s, ys, zs)ds.

Substitute forMt,we have

Yt=M0+

t

Z

0

Z_s^∗dWs−

t

Z

0

f(s, ys, zs)ds

=ξ+

T

Z

0

f(s, ys, zs)ds−

T

Z

0

Z_s^∗dWs+

t

Z

0

Z_s^∗dWs−

t

Z

0

f(s, ys, zs)ds

=ξ+

T

Z

t

f(s, ys, zs)ds−

T

Z

t

Z_s^∗dWs.

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SinceYtis adapted, we have Yt=E[Yt|Ft]

=E



ξ+

T

Z

t

f(s, ys, zs)ds−

T

Z

t

Z_s^∗dWs|Ft





=E



ξ+

T

Z

t

f(s, ys, zs)ds|Ft



−E





T

Z

t

Z_s^∗dWs|Ft





=E



ξ+

T

Z

t

f(s, ys, zs)ds|Ft



.

Y is square integrable sincef, ξ are square integrable. Let (y¹, z¹), and(y², z²)be two elements ofH^2,d

T,β ×H²

T,βR^n×d, and let(Y¹, Z¹)and(Y², Z²)be the associated solutions.

By Proposition 1applied with C = 0and β = µ²,we have(f, ξ)standard parameters, δ2fs = f(s, y_s¹, z_s¹)−f(s, y_s², z_s²)and δYT = 0.Then from Equations (9) and (10), we have

||δY||²_β ≤ T βE





T

Z

0

e^βs|f(s, y_s¹, z¹_s)−f(s, y²_s, z_s²)|²ds



, and

||δZ||²_β ≤ 1 βE





T

Z

0

e^βs|f(s, y_s¹, z_s¹)−f(s, y_s², z_s²)|²ds



.

Sincef is Lipschitz with constantCand using(a+b)² ≤2(a²+b²), we have

||δY||²_β +||δZ||²_β ≤ T

β + 1 β

CE





T

Z

0

e^βs(|δy|+|δz|)²ds





≤ T

β + 1 β

CE





T

Z

0

e^βs2(|δy|²+|δz|²)ds





= 2(1 +T)C β

||δy||²_β+||δz||²_β

. (14)

Choosingβ > 2(1+T)C, we see that this mapping is a contraction fromH^2,d

T,β×H²_T,βR^n×d onto itself and that there exists a fixed point, which is a unique continuous solution of the BSDE.

From Equation (14), we show that the Picard iterative sequence converges almost surely

Backward stochastic differential equations with applications