Backward Doubly Stochastic Equations with Jumps and Comparison Theorems

0.25 MB·

by  Wei Xu

BACKWARD DOUBLY STOCHASTIC EQUATIONS WITH
JUMPS AND COMPARISON THEOREMS
∗
Wei Xu
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1 Beijing Normal University
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2
January 19, 2016
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R Abstract
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In this work we mainly prove the existence and pathwise uniqueness of solutions to general
.
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backward doubly stochastic differential equations with jumps appearing in both forward and
t
a backward integral parts. Several comparison theorems under some weak conditions are also
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given. Finallyweapplycomparisontheoremsinprovingtheexistenceofsolutiontosomespecial
[ backward doubly stochastic differential equations with drift coefficient increasing linearly.

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v
7 1 Introduction
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2
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Backward stochastic differential equations (BSDEs) in the linear case were introduced in Kush-
0
. ner (1972), Bismut (1976), Bensoussan (1982) and Haussmann (1986) as adjoint processes in the
1
maximum principle for stochastic control problems and the pricing of options. Since the important
0
6 workof PardouxandPeng(1990), theinterestinBSDEshasincreased considerablyinrecent years.
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The significance of BSDEs is not only proved by the considerably important role they are playing
:
v in the study of partial differential equations (PDEs); see Peng (1991), Pardoux and Peng (1992)
i
X and Darling and Pardoux (1997), but also can be found in many other fields such as mathematical
r economics, financial mathematics, insurance and stochastic control. Here we just list several im-
a
portant works in every field. Duffie and Epstein (1992a,b) used BSDEs as a powerful tool to study
stochastic differential utility. Moreover, in the insurance market BSDEs are used in pricing and
hedging insurance equity-linked claims and asset-liability management problems, see Karoui et al.
(1997) and Delong (2013). Peng (1993) studied stochastic optimal control systems, where the state
variables are described by a system of ordinary-SDE and BSDEs, and derived a local form of the
maximum principle.
As further extensions of BSDEs, backward doubly stochastic differential equations (BDSDEs)
contain both forward and backward stochastic integrals. Those equations were first introduced
by Pardoux and Peng (1994) in the study of quasi-linear parabolic stochastic partial differential
equations (SPDEs). Compared to BSDEs, much less results about BDSDEs can be found in the
literature and most of the results established are about BDSDEs driven by Brownian motions. For
the details about applications of BDSDEs to SPDEs driven by Brownian motion, one can refer
†MSC2010 subject classifications: 60H05, 60H10, 37H10
†Keywords andphrases: Backwarddoublystochasticdifferentialequations,jump,comparisontheorem,Gaussian
white noise.
∗Supported byNSFC (No. 11401012).
1
to Zhang and Zhao (2007) which studied the existence and uniqueness of solution to BDSDEs on
infinite horizons, and the stationary solutions to SPDEs by virtue of the solutions to BDSDEs on
infinite horizons. Moreover, some work about BDSDEs with jumps appearing in the system of
ordinary-SDE have been published recently. For instance, Zhu and Shi (2012) studied BDSDEs
driven by Brownian motions and Poisson process with non-Lipschitz coefficients on random time
interval. Aman (2012), Aman and Owo (2012) and Ren et al. (2009) study a special reflected
generalized BDSDEs (driven by Teugel’s martingales associated with L´evy process) with means of
the penalization method and the fixed-point theorem. Existence and uniqueness of the solution
to the BDSDE with jumps in the forward integral are studies in Sow (2011) for the case of non-
Lipschitz coefficients. Recently, some results about stochastic control problems of BDSDEs have
been obtained by Han et al. (2010) and Bahlali and Gherbal (2010).
This work is motivated by Xiong (2013) and He et al. (2014) which mainly studied the distri-
bution function valued process of super-Brownian motions and super-L´evy processes characterized
as the pathwise unique solution to a SPDE. For any super-L´evy process with transition semigroup
(Q ) defined by (1.4) in He et al. (2014), they proved that its distribution function valued
t t≥0
process X(t) solve the following stochastic integral equation: for any x R,
∈
t t Xs−(x)
X (x) = X (x)+ A∗X (x)ds+√c W(ds,du)
t 0 s
Z0 Z0 Z0
t ∞ Xs−(x) t
+ zN˜(ds,dz,du) b X (x)ds, (1.1)
s
−
Z0 Z0 Z0 Z0
where W(ds,du);t 0,u > 0 is a Gaussian white noise with intensity dsπ(du), N(dt,dz,du) :
{ ≥ } {
t 0,u > 0 is a Poisson random measure with intensity dtµ(dz)du and A∗ is the dual operator of
≥ }
A defined by: for any f(x) C2(R),
∈ 0
1
Af(x)= βf′(x)+ σ2f′′(x)+ [f(x+z) f(x) f′(x)z1 ]ν(dz).
{|z|≤1}
2 R − −
Z
Furthermore, for any fixed T >0, define the Gaussian white noise WT(dt,du) : 0 t T,u E
{ ≤ ≤ ∈ }
by
WT([T t,T] A) = W([0,t] A), 0 t T,A B(E);
− × × ≤ ≤ ∈
and the Poisson random measures NT(dt,du) : 0 t T,u U by:
0
{ ≤ ≤ ∈ }
NT([T t,T] B)= N([0,t] B), 0 t T,B B(U ).
i
− × × ≤ ≤ ∈
In the proof of the pathwise uniqueness of solutions to (1.1), they established its connection to the
following BDSDE:
T T− XT−s(Lrs+x)
X (Lr +x) = X (Lr +x) b X (Lr +x)ds+√c W(ds,du)
T−t t 0 t − T−s s
Zt Zt− Z0
T− ∞ XT−s(Lrs+x) T
+ zN˜T(←d−s,dz,du) σ X (Lr +x)dB
0 − ∇ T−s s s
Zt− Z0 Z0 Zt
T
[X (Lr +x z) X (Lr +x)]M˜(ds,dz),
− ∇ T−s s − − T−s s
Zt
where T > 0 is fixed and Lr =L L , where L :t 0 is a L´evy process with generator A∗.
t t− r { t ≥ }
The purpose of this work is extending the above equations into more general BDSDEs with
jumpsappearingnotonlyintheforwardstochasticintegralpartbutalsointhebackwardstochastic
integral part; see(2.2)inSection 2. Pathwise uniquenessandexistence oftheir solutions areproved
inSection2and3respectivelyunderLipschitzconditions. Inaddition,severalcomparisontheorems
2
for BDSDEs will also be given in Section 4, since they play an important role in both theory and
applications; see Shi et al. (2005). Effected by random terms in the backward integrals, classical
methods are outof work, we useanother method to get comparison theorems with some reasonable
and weak conditions. The main difficulty is to deal with the influence of forward integrals to the
drift coefficient and backward integrals. As an applications of comparison theorems, in Section 5
we prove that solutions to a special kind of BDSDEs with drift coefficient increasing linearly exist.
Notation: For any n-dimensional vector X = (x , ,x ), Y = (y , ,y ) and n n-
1 n 1 n
matrix A = (a ), let X 2 = n x2, X,Y = n x·y··, T(A) = (a , ··,·a ) and A×2 =
Tr(ATA) = ni,j a2 ,kwhkere Tr(Ai=)1is ithehtracei of A.i=F1orianiy f C2(Rn1)1, l·e·t· nn k k
i,j=1 ij P P ∈
P ∂f(x) ∂f(x) ∂2f(x) ∂2f(x)
Df(x)= , , and D2f(x) = , , .
∂x ··· ∂x ∂x2 ··· ∂x2
1 n 1 n
(cid:16) (cid:17) (cid:16) (cid:17)
Throughout this paper, we make the conventions
b ∞ b−
= , = and =
Za Z(a,b] Za Z(a,∞) Za− Z[a,b)
for any b a 0.
≥ ≥
2 Pathwise Uniqueness
In this section, we mainly study the pathwise uniqueness of solutions to general backward doubly-
stochastic equations. Suppose that T > 0 is a fixed constant and (Ω,F,P) is a complete prob-
ability space endowed with filtration G(1) satisfying the usual hypotheses. Let B(s) is a
{ t }0≤t≤T
n-dimensional(G(1))-Brownianmotionand M(dt,du) :0 t T,u F a(G(1))-Poissonrandom
t { ≤ ≤ ∈ } t
measurewithintensitydtν(du),whereν(du)isaσ-finiteBorelmeasureonthePolishspacesF. Oth-
erwise, let G(2) beanother filtration on (Ω,F,P) satisfying the usual hypotheses and inde-
{ t }0≤t≤T
pendent with G(1) . Let W(ds,du) = (W (ds,du), ,W (ds,du))T;0 t T,u E be
{ t }0≤t≤T { 1 ··· n ≤ ≤ ∈ }
an-dimensional(G(2))-Gaussian whitenoise constructed with northogonal whitenoises W (ds,du)
t i
on R+ E with intensity dsπ (du) respectively. Here we denote π(du) = (π (du), ,π (du))T.
i 1 n
× ···
Suppose µ (du) is a σ-finite Borel measure on the Polish space U and µ (du) is a finite Borel
0 0 1
measure on the Polish space U . Moreover, For each i = 0,1, let N (dt,du) : 0 t T,u U
1 i i
{ ≤ ≤ ∈ }
be a (G(2))-Poisson random measure with intensity dtµ (du). Obviously, all the random elements
t i
introduced above are independent of each other.
Denote Gr = σ(G(1) G(2) ) for 0 r t T. Specially, G0 = σ(G(1) G(2)) and GT−t =
t t ∪ T−r ≤ ≤ ≤ t t ∪ T T
σ(G(1) G(2)) are two filtrations satisfying the usual hypotheses. It is easily seen that B is
T ∪ t { t}
a (G0)-Brownian motion and M(dt,du) is a (G0)-Poisson random measure. Define the Gaussian
t t
white noise WT(dt,du) : 0 t T,u E by
{ ≤ ≤ ∈ }
WT([T t,T] A) = W([0,t] A), 0 t T,A B(E).
− × × ≤ ≤ ∈
For i = 0,1, define the Poisson random measures NT(dt,du) :0 t T,u U by:
{ i ≤ ≤ ∈ i}
NT([T t,T] B)= N ([0,t] B), 0 t T,B B(U ).
i − × i × ≤ ≤ ∈ i
Roughly speaking, we can consider WT(dt,du) and NT(dt,du) as the time reversal of W(dt,du)
i
and N (dt,du), respectively.
i
3
A real process ξ is said to be (Gr)-progressive if for any 0 r t T, the mapping
{ s}0≤s≤T t ≤ ≤ ≤
(s,ω) ξ (ω) restricted to [r,t] Ω is B([r,t]) Gr-measurable. A two-parameter real process
7→ s × × t
ζ (u) is said to be (Gr)-progressive if for any 0 r t T, the restriction of
{ s }0≤s≤T,u∈E t ≤ ≤ ≤
(s,u,ω) ζ (u,ω) to [r,t] E Ω is B([r,t]) B(E) Gr-measurable.
7→ s × × × × t
Let P denote the σ-algebra on Ω [0,T] generated by all real-valued left continuous pro-
×
cesses which are (Gr)-progressive. A process (ξ ) is said to be predictable if the mapping
t s 0≤s≤T
(ω,s) ξ (ω) is P-measurable. Also a two-parameter process ζ (u) is said to be
s s 0≤s≤T,u∈E
7→ { }
predictable if the mapping (ω,s,x) ζ (ω,x) is (P B(E))-measurable. For the theory of time-
s
7→ ×
space stochastic integrals of predictable two parameter processes with respect to point processes or
random measures, readers can refer to Section II.3 in Ikeda and Watanabe (1989). The stochastic
integrals with respect to martingale measures were discussed in Section 7.3 of Li (2011). We make
the convention that the stochastic integral of a progressive process refers to a predictable version
of the integrand. The existence of such a version was briefly discussed in Section 2 of He et al.
(2014). To simplify the following statements, we introduce several Banach spaces:
(1) S2 := (ξ ) :ξ is (Gr)-progressive and ξ < , where
G,T { s 0≤s≤T s t k kS2T ∞}
ξ 2 = E sup ξ 2 .
S2 s
k k T s∈[0,T]k k
h i
(2) L2 := (β ) : β is (Gr)-progressive and β < , where
G,T s 0≤s≤T s t k kLT2 ∞
n o
T
β 2 =E β 2ds .
L2 s
k k T (cid:26)Z0 k k (cid:27)
(3) L2 (E) := σ(s,u) : σ(s,u) is (Gr)-progressive and σ < , where
G,T { }0≤s≤T,u∈E t k kLT2(E) ∞
n o
T T
σ 2 = E σ(s, ) 2 ds = E ds T(σT(s,u)σ(s,u))π(du) .
k kLT2(E) (cid:26)Z0 k · kL2(E) (cid:27) (cid:26)Z0 ZE (cid:27)
(4) L2 (U ) := g(s,u) :g(s,u) is (Gr)-progressive and g < , where
G,T 0 { }0≤s≤T,u∈U0 t k kLT2(U0) ∞
n o
T T
g 2 = E g(s, ) 2 ds = E ds g(s,u) 2µ (du) .
k kLT2(U0) (cid:26)Z0 k · kL2(U0) (cid:27) (cid:26)Z0 ZU0k k 0 (cid:27)
(5) L2 (U ) := g(s,u) :g(s,u) is (Gr)-progressive and g < , where
G,T 1 { }0≤s≤T,u∈U1 t k kLT2(U1) ∞
n o
T T
g 2 = E g(s, ) 2 ds = E ds g(s,u) 2µ (du) .
k kLT2(U1) (cid:26)Z0 k · kL2(U1) (cid:27) (cid:26)Z0 ZU1k k 1 (cid:27)
(6) L2 (F) := ζ (u) : ζ (u) is (Gr)-progressive and ζ < , where
G,T s s t k kLT2(F) ∞
n o
T T
ζ 2 = E ζ 2 ds = E ds ζ (u) 2ν(du) .
k kLT2(F) (cid:26)Z0 k skL2(F) (cid:27) (cid:26)Z0 ZF k s k (cid:27)
Before giving the main results, we extend Itˆo formula to the general case. Let X be a m-
t
dimensional stochastic process defined by:
T T T−
X = X + b(s)ds+ a(s,u)WT(←d−s,du)+ γ (s,u)N˜T(←d−s,du)
t T 0 0
Zt Zt ZE Zt− ZU0
4
T− T T
+ γ (s,u)NT(←d−s,du) Z dB(s) ζ (u)M˜(ds,du), (2.1)
1 1 − s − s
Zt− ZU1 Zt Zt ZF
whereb(s),a(s,u), γ (s,u), γ (s,u)andζ (u)arem-dimensional(Gr)-progressiveprocesses,a(s,u)
0 1 s t
and Z are (Gr)-progressive (m n)-matrix-valued processes.
s t ×
Proposition 2.1 For any f C2(Rm,R), we have
∈
T T
f(X ) = f(X )+ Df(X )b (s)ds+ Df(X )a(s,u)WT(←d−s,du)
t T s i s
Zt Zt ZE
T
+ ds T[aT(s,u)D2f(X )a(s,u)]π(du)
s
Zt ZE
T
+ [f(X +γ (s,u)) f(X )]N˜T(←d−s,du)
s 0 − s 0
Zt ZU0
T
+ ds [f(X +γ (s,u)) f(X ) Df(X )γ (s,u)]µ (du)
s 0 s s 0 0
− −
Zt ZU0
T
+ [f(X +γ (s,u)) f(X )]NT(←d−s,du)
s 1 − s 1
Zt ZU1
T 1 T
Df(X )Z dB(s) Tr(ZT(s)D2f(X )Z )ds
s s s s
− − 2
Zt Zt
T
[f(X +ζ (u)) f(X )]M˜(ds,du)
s s s
− −
Zt ZF
T
ds [f(X +ζ (u)) f(X ) Df(X )ζ (u)]ν(du).
s s s s s
− − −
Zt ZF
Remark 2.2 As in He et al. (2014), we make the convention that the stochastic integral of a
progressive process always refers to that of a predictable version of the integrand. Here we emphasis
that the integrals in (2.2) denote by WT(←d−s,du), N˜T(←d−s,du) and NT(←d−s,du) are backward ones,
0 1
which can be defined as the time-reversal of the corresponding forward stochastic integrals; see He
et al. (2014) for more precise explanations. Of course, the integrals with respect to dB(s) and
M˜(ds,du) in (2.2) are forward ones.
Now let us introduce the backward doubly stochastic integral equation to work with. Suppose
that we have the following measurable mappings:
β : [0,T] Rm Rm×n L2 (F) Rm;
× × × G,T 7→
σ : [0,T] Rm Rm×n L2 (F) E Rm×n;
× × × G,T × 7→
g : [0,T] Rm Rm×n L2 (F) U Rm;
0 × × × G,T × 0 7→
g : [0,T] Rm Rm×n L2 (F) U Rm.
1 × × × G,T × 1 7→
Given Y G0, we consider the equation:
T ∈ T
T T
Y = Y + β(s,Y ,Z ,ζ )ds+ σ(s,Y ,Z ,ζ ,u)WT(←d−s,du)
t T s s s s s s
Zt Zt ZE
T− T−
+ g (s,Y ,Z ,ζ ,u)N˜T(←d−s,du)+ g (s,Y ,Z ,ζ ,u)NT(←d−s,du)
0 s s s 0 1 s s s 1
Zt− ZU0 Zt− ZU1
T T
Z dB ζ (u)M˜(ds,du). (2.2)
s s s
− −
Zt Zt ZF
Definition 2.3 We call the process (Y ,Z ,ζ (u)) a solution to (2.2) if it is (Gr)-progressive
t t t 0≤t≤T t
and for any 0 r t T the equation (2.2) is satisfied almost surely.
≤ ≤ ≤
5
Condition 2.4 There exist constants C > 0 and 0 < α < 1 such that for any s [0,T] and
∈
(x ,y ,z ,ζ ) Rm Rm Rm×n L2 (F) with i= 1,2,
i i i i ∈ × × × G,T
β(s,y ,z ,ζ ) β(s,y ,z ,ζ ) 2 C y y 2+ z z 2+ ζ ζ 2 (2.3)
k 1 1 1 − 2 2 2 k ≤ k 1− 2k k 1 − 2k k 1− 2kL2(F)
and (cid:0) (cid:1)
σ(s,y ,z ,ζ , ) σ(s,y ,z ,ζ , ) 2
k 1 1 1 · − 2 2 2 · kL2(E)
+ g (s,y ,z ,ζ , ) g (s,y ,z ,ζ , ) 2
k 0 1 1 1 · − 0 2 2 2 · kL2(U0)
+ g (s,y ,z ,ζ , ) g (s,y ,z ,ζ , ) 2
k 1 1 1 1 · − 1 2 2 2 · kL2(U1)
C y y 2+α z z 2+α ζ ζ 2 . (2.4)
≤ k 1− 2k k 1 − 2k k 1− 2kL2(F)
(1) (1) (1) (2) (2) (2)
Theorem 2.5 Suppose Condition 2.4 holds. If (Y ,Z ,ζ (u)) and (Y ,Z ,ζ (u)) are
t t t t t t
(1) (2)
solutions to (2.2) with Y = Y a.s., then
T T
(1) (2)
P Y = Y for all t [0,T] = 1 (2.5)
t t ∈
(cid:16) (cid:17)
and
Z(1) Z(2) + ζ(1) ζ(2) = 0. (2.6)
k − kLT2 k − kLT2(F)
Proof. Let (Y¯,Z¯ ,ζ¯(u)) = (Y(1) Y(2),Z(1) Z(2),ζ(1)(u) ζ(2)(u)). From (2.2) we get
t t t t − t t − t t − t
T T T−
Y¯ = β¯(s)ds+ σ¯(s,u)WT(←d−s,du)+ g¯(s,u)N˜T(←d−s,du)
t 0 0
Zt Zt ZE Zt− ZU0
T− T T
+ g¯(s,u)NT(←d−s,du) Z¯ dB ζ¯(u)M˜(ds,du), (2.7)
1 1 − s s − s
Zt− ZU1 Zt Zt ZF
where
β¯(s) = β(s,Y(1),Z(1),ζ(1)) β(s,Y(2),Z(2),ζ(2)),
t t t − t t t
(1) (1) (1) (2) (2) (2)
σ¯(s,u) = σ(s,Y ,Z ,ζ ,u) σ(s,Y ,Z ,ζ ,u),
t t t − t t t
(1) (1) (1) (2) (2) (2)
g¯(s,u) = g (s,Y ,Z ,ζ ,u) g (s,Y ,Z ,ζ ,u),
0 0 t t t − 0 t t t
(1) (1) (1) (2) (2) (2)
g¯(s,u) = g (s,Y ,Z ,ζ ,u) g (s,Y ,Z ,ζ ,u).
1 1 t t t − 1 t t t
By Proposition 2.1, we have
T T T
Y¯ 2 = 2 Y¯ ,β¯(s) ds+2 Y¯ ,σ¯(s,u) WT(←d−s,du)+ σ¯(s, ) 2 ds
k tk h s i h s i k · kL2(E)
Zt Zt ZE Zt
T T
+ [2 Y¯ ,g¯(s,u) + g¯(s,u) 2]N˜T(←d−s,du)+ g¯(s, ) 2 ds
h s 0 i k 0 k 0 k 0 · kL2(U0)
Zt ZU0 Zt
T T
+2 Y¯ ,g¯(s,u) NT(←d−s,du)+ g¯(s,u) 2NT(←d−s,du)
h s 1 i 1 k 1 k 1
Zt ZU1 Zt ZU1
T T T
2 Y¯ ,Z¯ dB Z¯ 2ds ζ¯ 2 ds
− h s si s − k sk − k skL2(F)
Zt Zt Zt
T
[2 Y¯ ,ζ¯(u) + ζ¯(u) 2]M˜(ds,du).
s s s
− h i k k
Zt ZF
From Cauchy’s inequality, for any a,b > 0 we have
T T
E Y¯ 2 +E Z¯ 2ds +E ζ¯ 2 ds
k tk k sk k skL2(F)
hZt i hZt i
(cid:2) (cid:3)
6
T T T
= E 2 Y¯ ,β¯(s) ds +E σ¯(s, ) 2 ds +E g¯(s, ) 2 ds
h s i k · kL2(E) k 0 · kL2(U0)
(cid:26) Zt (cid:27) (cid:26)Zt (cid:27) (cid:26)Zt (cid:27)
T T
+E 2 ds Y¯ ,g¯(s,u) µ (du) +E g¯(s, ) 2 ds
h s 1 i 1 k 1 · kL2(U1)
(cid:26) Zt ZU1 (cid:27) (cid:26)Zt (cid:27)
1 µ (U ) T T T
( + 1 1 )E Y¯ 2ds +aE β¯(s) 2ds +E σ¯(s, ) 2 ds
≤ a b k sk k k k · kL2(E)
(cid:26)Zt (cid:27) (cid:26)Zt (cid:27) (cid:26)Zt (cid:27)
T T
+E g¯(s, ) 2 ds +(1+b)E g¯(s, ) 2 ds .
k 0 · kL2(U0) k 1 · kL2(U1)
(cid:26)Zt (cid:27) (cid:26)Zt (cid:27)
Since µ is a finite measure, by H¨older’s inequality and Condition 2.4,
1
T T
E Y¯ 2 +E Z¯ 2ds +E ζ¯ 2 ds
k tk k sk k skL2(F)
(cid:2) 1 (cid:3) µ (hUZ)t T i hZt T i
+ 1 1 E Y¯ 2ds +CaE [ Y¯ 2+ Z¯ 2+ ζ¯ 2 ]ds
≤ a b k sk k sk k sk k skL2(F)
(cid:16) (cid:17)T (cid:26)Zt (cid:27) (cid:26)Zt (cid:27)
+(1+b)E [C Y¯ 2+α Z¯ 2+α ζ¯ 2 ]ds .
k sk k sk k skL2(F)
(cid:26)Zt (cid:27)
Here we can choose a,b small enough such that αˆ := Ca+α+bα< 1. Then
T T
E Y¯ 2 +(1 αˆ)E Z¯ 2ds +(1 αˆ)E ζ¯ 2 ds
k tk − k sk − k skL2(F)
(cid:26)Zt (cid:27) (cid:26)Zt (cid:27)
(cid:2) (cid:3) T
1/a+1/b+C(1+a+b) E Y¯ 2ds .
s
≤ k k
(cid:26)Zt (cid:27)
(cid:2) (cid:3)
By Gronwall’s lemma, we have
T T
E Y¯ 2 +E Z¯ 2ds +E ζ¯ 2 ds = 0.
k tk k sk k skL2(F)
(cid:26)Zt (cid:27) (cid:26)Zt (cid:27)
(cid:2) (cid:3)
This implies (2.6). Then for any fixed t [0,T], the six terms on the right-hand sideof (2.7) vanish
∈
almost surely. Since each of the six terms is right-continuous or left-continuous, they almost surely
vanish for all t [0,T]. Thus we have finished the proof. (cid:3)
∈
3 Existence
In this section, we study the existence of solutions to (2.2). For any 0 r t T we define the
≤ ≤ ≤
natural σ-algebras:
FBM = σ( B(s) B(r),M((r,s] A) :r s t,A B(F) ) N ,
r,t { − × ≤ ≤ ∈ } ∨
FWN = σ( W((r,s] A),N ((r,s] B),N ((r,s] C):
r,t { × 0 × 1 ×
r s t,A B(E),B B(U ),C B(U ) ) N ,
0 1
≤ ≤ ∈ ∈ ∈ } ∨
where N denotes the totality of P-null sets. For simplicity, we write FBM = FBM and FWN =
t 0,t t
FWN. Let Fr = FBM FWN for 0 r t T. Similarly, we can defineS2 , L2 , L2 (E),
0,t t t ∨ T−r ≤ ≤ ≤ F,T F,T F,T
L2 (U ), L2 (U ), L2 (F) like those in the last section but with Gr : 0 r t T
F,T 0 F,T 1 F,T { t ≤ ≤ ≤ }
replaced by Fr :0 r t T .
{ t ≤ ≤ ≤ }
Theorem 3.1 Suppose Condition 2.4 holds. Then there exists a solution (Y ,Z ,ζ (u)) to (2.2) in
t t t
S2 L2 L2 (F).
F,T × F,T × F,T
7
Obviously, combining this theorem with Theorem 2.5, we have solution to (2.2) exists uniquely
in S2 L2 L2 (F). Before giving the proof of Theorem 3.1, we introduce a lemma about
G,T × G,T × G,T
the solution to some simple backward doubly stochastic equation, which is very important in the
proof of this theorem.
Lemma 3.2 Let β L1 , σ L2 (E), g L2 (U ) and g L2 (U ). Then for any
∈ F,T ∈ F,T 0 ∈ F,T 0 1 ∈ F,T 1
Y F0 with finite second moment, there exists a unique solution (Y ,Z ,ζ (u)) S2 L2
T ∈ T t t t ∈ F,T × F,T ×
L2 (F) to the following equation:
F,T
T T T−
Y = Y + β(s)ds+ σ(s,u)WT(←d−s,du)+ g (s,u)N˜T(←d−s,du)
t T 0 0
Zt Zt ZE Zt− ZU0
T− T T
+ g (s,u)NT(←d−s,du) Z dB ζ (u)M˜(ds,du). (3.1)
1 1 − s s− s
Zt− ZU1 Zt Zt ZF
Proof. The uniqueness of the solution follows from Theorem 2.5. Recall F0 = FBM FWN for
t t ∨ T
0 t T. Observe that
≤ ≤
T T−
Ψ := Y + σ(s,u)WT(←d−s,du)+ g (s,u)N˜T(←d−s,du)
T T 0 0
Z0 ZE Z0− ZU0
T T−
+ β(s)ds+ g (s,u)NT(←d−s,du) (3.2)
1 1
Z0 Z0− ZU1
is F0-measurable. Then we can define a Doob’s martingale:
T
M = E[Ψ F0], 0 t T.
t T| t ≤ ≤
Since Ft F0, from (3.2) we have
t ⊂ t
t t
M = Y + β(s)ds+ σ(s,u)WT(←d−s,du)
t t
Z0 Z0 ZE
t− t−
+ g (s,u)N˜T(←d−s,du)+ g (s,u)NT(←d−s,du), (3.3)
0 0 1 1
Z0− ZU0 Z0− ZU1
where Y = E[Ξ(t)F0] and
t | t
T T
Ξ(t) = Y + β(s)ds+ σ(s,u)WT(←d−s,du)
T
Zt Zt ZE
T− T−
+ g (s,u)N˜T(←d−s,du)+ g (s,u)NT(←d−s,du). (3.4)
0 0 1 1
Zt− ZU0 Zt− ZU1
By the martingale representation theorem, see Lemma 2.3 in Tang and Li (1994), there exist
(F0)-progressive processes Z and ζ (u) such that
t { s} { s }
t t
M = M + Z dB + ζ (du)M˜(ds,du)
t 0 s s s
Z0 Z0 ZF
and hence
T T
M = M + Z dB + ζ (u)M˜(ds,du). (3.5)
T t s s s
Zt Zt ZF
Since M = Ψ , we can substitute (3.2) and (3.3) into (3.5) to obtain (3.1). Finally, we need to
T T
prove for any 0 r T the process (Y ,Z ,ζ (u)) is (Fr)-progressive. Observe that
≤ ≤ t t t r≤t≤T,u∈F t
Y = E[Ξ(r)F0]= E[Ξ(r)FBM FWN]= E[Ξ(r)Fr FWN ],
r | r | r ∨ T | r ∨ T−r,T
8
where FWN and FWN are independent. By (3.4) it is easy to see that Ξ(r) is independent of
T−r T−r,T
FWN . Then we have Y = E[Ξ(r)Fr], which is Fr-measurable. By (3.1) we have
T−r,T r | r r
T T T T−
Z dB + ζ (u)M˜(ds,du) = σ(s,u)WT(←d−s,du)+ g (s,u)N˜T(←d−s,du)
s s s 0 0
Zr Zr ZF Zr ZE Zr− ZU0
T T−
+Y Y + β(s)ds+ g (s,u)NT(←d−s,du).
T − r 1 1
Zr Zr− ZU1
Then by the uniqueness of the martingale representation, the process (Z ,ζ (u)) has an (Fr)-
t t t
progressive version. Since each term in (3.1) is right or left continuous, the process (Y ) is (Fr)-
t t
progressive. (cid:3)
Proof of Theorem 3.1. We shall use a Picard iteration argument to construct a solution to (2.2).
(0) (0) (0)
Let Y = Z = ζ (u) 0. By Lemma 3.2, for any n 0 there exists a unique solution
t t t ≡ ≥
(n+1) (n+1) (n+1)
(Y ,Z ,ζ (u)) to the following equation:
t t t
T T
Y(n+1) = Y + β(s,Y(n),Z(n),ζ(n))ds+ σ(s,Y(n),Z(n),ζ(n),u)WT(←d−s,du)
t T s s s s s s
Zt Zt ZE
T− T
+ g (s,Y(n),Z(n),ζ(n),u)N˜T(←d−s,du) Z(n+1)dB
0 s s s 0 − s s
Zt− ZU0 Zt
T− T
+ g (s,Y(n),Z(n),ζ(n),u)NT(←d−s,du) ζ(n+1)(u)M˜(ds,du).
1 s s s 1 − s
Zt− ZU1 Zt ZF
Let Y¯(n+1) = Y(n+1) Y(n), Z¯(n+1) = Z(n+1) Z(n) and ζ¯(n+1)(u) = ζ(n+1)(u) ζ(n)(u). From
t t − t t t − t t t − t
(2.2) we have
T T T−
Y¯(n+1) = β¯(n)(s)ds+ σ¯(n)(s,u)WT(←d−s,du)+ g¯(n)(s,u)N˜T(←d−s,du)
t 0 0
Zt Zt ZE Zt− ZU0
T− T T
+ g¯(n)(s,u)NT(←d−s,du) Z¯(n+1)dB ζ¯(n+1)(u)M˜(ds,du),
1 1 − s s− s
Zt− ZU1 Zt Zt ZF
where
β¯(n)(s) = β(s,Y(n),Z(n),ζ(n)) β(s,Y(n−1),Z(n−1),ζ(n−1)),
s s s − s s s
σ¯(n)(s,u) = σ(s,Y(n),Z(n),ζ(n),u) σ(s,Y(n−1),Z(n−1),ζ(n−1),u),
s s s − s s s
g¯(n)(s,u) = g (s,Y(n),Z(n),ζ(n),u) g (s,Y(n−1),Z(n−1),ζ(n−1),u),
0 0 s s s − 0 s s s
g¯(n)(s,u) = g (s,Y(n),Z(n),ζ(n),u) g (s,Y(n−1),Z(n−1),ζ(n−1),u).
1 1 s s s − 1 s s s
According to Proposition 2.1,, we have
T T
Y¯(n+1) 2 = 2 Y¯(n+1),β¯(n)(s) ds+2 Y¯(n+1),σ¯(n)(s,u) WT(←d−s,du)
k t k h s i h s i
Zt Zt ZE
T T
+ σ¯(n)(s, ) 2 ds+ g¯(n)(s, ) 2 ds
k · kL2(E) k 0 · kL2(U0)
Zt Zt
T−
+ [2 Y¯(n+1),g¯(n)(s,u) + g¯(n)(s,u) 2]N˜T(←d−s,du)
h s 0 i k 0 k 0
Zt− ZU0
T−
+ [2 Y¯(n+1),g¯(n)(s,u) + g¯(n)(s,u) 2]NT(←d−s,du)
h s 1 i k 1 k 1
Zt− ZU1
T T T
2 Y¯(n+1),Z¯(n+1) dB Z¯(n+1) 2ds ζ¯(n+1) 2 ds
− h s s i s − k s k − k s kL2(F)
Zt Zt Zt
T
[2 Y¯(n+1),ζ¯(n+1)(u) + ζ¯(n+1)(u) 2]M˜(ds,du).
− h s s i k s k
Zt ZF
9
It follows that
T T
E Y¯(n+1) 2 +E Z¯(n+1) 2ds +E ζ¯(n+1) 2 ds
| t | k s k k s kL2(F)
(cid:26)Zt (cid:27) (cid:26)Zt (cid:27)
(cid:2) (cid:3) T T
= E 2 Y¯(n+1),β¯(n)(s) ds +E σ¯(n)(s, ) 2 ds
h s i k · kL2(E)
(cid:26) Zt (cid:27) (cid:26)Zt (cid:27)
T T
+E g¯(n)(s, ) 2 ds +E g¯(n)(s, ) 2 ds
k 0 · kL2(U0) k 1 · kL2(U1)
(cid:26)Zt (cid:27) (cid:26)Zt (cid:27)
T
+E 2 ds Y¯(n+1),g¯(n)(s,u) µ (du) .
h s 1 i 1
(cid:26) Zt ZU1 (cid:27)
By integration by parts, one can see, for any λ > 0,
T T
λ eλsE Y¯(n+1) 2 ds = eλsE Y¯(n+1) 2 T eλsdE[ Y¯(n+1) 2]
k s k k s k t − k s k
Zt Zt
(cid:2) (cid:3) (cid:2) (cid:3)(cid:12) T
= eλtE Y¯(n+1) 2(cid:12) +E 2 eλs Y¯(n+1),β¯(n)(s) ds
− k t k h s i
(cid:26) Zt (cid:27)
(cid:2)T (cid:3)
+E eλs σ¯(n)(s, ) 2 ds
k · kL2(E)
(cid:26)Zt (cid:27)
T
+E eλs g¯(n)(s, ) 2 ds
k 0 · kL2(U0)
(cid:26)Zt (cid:27)
T
+E eλs g¯(n)(s, ) 2 ds
k 1 · kL2(U1)
(cid:26)Zt (cid:27)
T
+E 2 eλsds Y¯(n+1),g¯(n)(s,u) µ (du)
h s 1 i 1
(cid:26) Zt ZU1 (cid:27)
T T
E Z¯(n+1) 2eλsds E eλs ζ¯(n+1) 2 ds .
− k s k − k s kL2(F)
(cid:26)Zt (cid:27) (cid:26)Zt (cid:27)
By H¨older’s inequality, for any a,b > 0 we have
T T T
λ eλsE Y¯(n+1) 2 ds+E eλs Z¯(n+1) 2ds +E eλs ζ¯(n+1) 2 ds
k s k k s k k s kL2(F)
Zt (cid:26)Zt (cid:27) (cid:26)Zt (cid:27)
(cid:2) (cid:3)T T
(1/a+1/b) eλsE Y¯(n+1) 2 ds+E a eλs β¯(n)(s) 2ds
≤ k s k k k
Zt (cid:26) Zt (cid:27)
T (cid:2) (cid:3)
+E eλs σ¯(n)(s,u) 2 + g¯(n)(s,u) 2 ds
k kL2(E) k 0 kL2(U0)
(cid:26)Zt (cid:27)
(cid:2) T (cid:3)
+(1+b)E eλs g¯(n)(s,u) 2 ds .
k 1 kL2(U1)
(cid:26)Zt (cid:27)
Using Condition 2.4, we have
T T T
λ eλsE Y¯(n+1) 2 ds+E eλs Z¯(n+1) 2ds +E eλs ζ¯(n+1) 2 ds
k s k k s k k s kL2(F)
Zt (cid:26)Zt (cid:27) (cid:26)Zt (cid:27)
(cid:2) T (cid:3) T
(1/a+1/b) eλsE Y¯(n+1) 2 ds+ aCE eλs Y¯(n) 2+ Z¯(n) 2+ ζ¯(n) 2 ds
≤ k s k k s k k s k k s kL2(F)
Zt (cid:26)Zt (cid:27)
T (cid:2) (cid:3) (cid:2) (cid:3)
+(1+b)E eλs C Y¯(n) 2+α Z¯(n) 2+α ζ¯(n) 2 ds .
k s k k s k k s kL2(F)
(cid:26)Zt (cid:27)
(cid:2) (cid:3)
Then
T
E eλs (λ 1/a 1/b) Y¯(n+1) 2+ Z¯(n+1) 2 + ζ¯(n+1) 2 ds
− − k s k k s k k s kL2(F)
(cid:26)Zt (cid:27)
(cid:2) (cid:3)
10