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 par-tial differenpar-tial 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.
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 ⇒Fc ∈ F whereFcis the complement ofF inΩandFc = 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.
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 inRd is a stochastic process having stationary and independent increments and we always assumeX0 = 0with probability1.So
• Xt: Ω−→Rd.
• 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 Rd 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].
Definition 5(Characteristic function). LetXbe a random variable defined on the proba-bility space in Definition1taking values inRdwith the probability lawPx.It’s character-istic functionφx :Rd−→Cis
φx(u) =E ei(u,x)
= Z
Ω
ei(u,X(ω))P(dω)
= Z
Rd
ei(u,y)Px(dy)
for eachu∈Rd
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∈Rd
E(ei(u,Xt)) =etη(u) where
η(u) =i(b, u)− 1
2(u, au) + Z
Rd−{0}
ei(u,y)−1−i(u, y)χ0<|y|<1(y)
ν(dy).
where
• b∈Rd
• ais a positive definite symmetricd×dmatrix
• νis a L´evy measure onRd− {0}so that Z
Rd−{0}
min{1,|y|2ν(dy)}<∞.
A L´evy process can be decomposed into a linear drift, Brownian motion, and a pure jump process [16]. This result is called the L´evy-Itˆo decomposition and defined as follows.
Theorem 2.1.1(The L´evy-Itˆo decomposition). IfX is a L´evy process, then there exists b ∈ Rd,a Brownian motionB with diffusion matrixQand an independent Poisson ran-dom measureN onR+×(Rd− {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)|+σ2ij(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.
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
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 anLp space before the H¨older inequality as follows.
Definition 9. Consider the measurable space in Definition 1 and 1 ≤ p, q ≤ ∞. The spaceLp(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].
TheLp norm off ∈L(Ω)is defined by
||f||Lp = Z
|f|pP(dω∈ F) 1p
. Whenp= 1the spaceL1consists of all integrable functions onΩ.
Theorem 2.1.5(H¨older inequality). Assume that a functionf ∈ Lp andg ∈ Lq,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).
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
tk≤t
(Xtk+1(ω)−Xtk(ω))(Ytk+1(ω)−Ytk(ω)).
[15].
Let us consider the Itˆo formula for Brownian motion and L´evy process driven Itˆo process as follows.
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, and are continuous. LetXtandYtbe Itˆo processes. The two dimensional Itˆo formula in differential form is 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) = semi-martingale whose every path is right-continuous. Letα < β be real numbers, and[0, T] is a sub interval of[0,∞).Then
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)eR0th(s)ds. In its simplest form wheng =Aandh=B are constants,
f(t)≤AeBt.
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)
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, Hta =H(a, t, ω)be a boundedA ⊗Pmeasurable function, and letµbe a finite measure on A. Let Zta = Rt
0 HsadXs be A ⊗ B(R+)⊗ F measurable such that for each a, Za is c´adl´ag version of Ha·X.Then Yt = R
AZtaµ(da)is c´adl´ag 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
Xs2ds
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, Xt2, Yt2)−f2(t, Xt2, Yt2)≥0, dPN dt a.s.
Then we have almost surely for any timet,
Xt1 ≥Xt2.
Moreover, the comparison is strict. That is, if, in addition, X01 = X02, then ξ1 = ξ2, f1(t, Xt2, Yt2) = f2(t, Xt2, Yt2), dPN
dt a.s., and X1 = X2 a.s. More generally,
ifXt1 =Xt2on a setA∈ Ft, thenXs1 =Xs2almost surely on[0, T]×A, ξ1 =ξ2a.s. on A, andf1(t, Xt2, Yt2) = f2(t, Xt2, Yt2)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(xt+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.