高维倒向重随机微分方程的比较定理及其应用
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∗ Partially supported by Foundation for University Key Teacher by Ministry of Education of China, National Natural Science Foundation of China grant 10201018 and Doctor Promotional Foundation of Shandong grant 02BS127.
(H2) There exist constants C > 0 and 0 < α < 1 such that for any (ω, t) ∈ Ω × [0, T ], (y1 , z1 ), (y2 , z2 ) ∈ Rk × Rk×d , |f (t, y1 , z1 ) − f (t, y2 , z2 )|2 ≤ C (|y1 − y2 |2 + z1 − z2 2 ) g(t, y1 , z1 ) − g(t, y2 , z2 ) 2 ≤ C |y1 − y2 |2 + α z1 − z2 2 Given ξ ∈ L2 (Ω, FT , P ; Rk ), we consider the following BDSDE:
Comparison Theorems of the multi-dimensional BDSDEs and Applico Shi Yufeng Shandong Art and Design School of Mathematics and School of Mathematics and Academy Jinan 250014, China Statistics Shandong Economic System Sciences Shandong University, Jinan 250100, China University, Jinan 250100, China E-mail:byhan01@ zhubo207@ yfshi@
T T T
yt = ξ +
t
f (s, ys , zs )ds +
t
g(s, ys , zs )dBs −
t
zs dWs .
(1)
Pardoux and Peng [6] showed that BDSDEs can produce a probabilistic representation for certain quasilinear stochastic partial differential equations (SPDEs).recently Shi, Gu and Liu[7] proved the comparison theorem of these BDSDEs, but they studied the one-dimensional BDSDEs. In this paper, we applying the idea of [5]zhou, prove the comparison theorem for multi-dimensional BDSDEs. Then we also study the multi-dimension BDSDEs with continuous coefficients as an application of the comparison theorem of multi-dimension BDSDEs.
η η where for any process {ηt }, Fs,t = σ {ηr − ηs ; s ≤ r ≤ t} ∨ N , Ftη = F0 ,t . We note that the collection {Ft ; t ∈ [0, T ]} is neither increasing nor decreasing, so it does not constitute a classical filtration. For any n ∈ N , let M 2 (0, T ; Rn ) denote the set of (classes of dP ⊗ dt a.e. equal) n-dimensional jointly measurable stochastic processes {ϕt ; t ∈ [0, T ]} which satisfy: T 2 (i) ϕ 2 M 2 := E 0 |ϕt | dt < ∞; (ii) ϕt is Ft -measurable, for any t ∈ [0, T ]. Similarly, we denote by S 2 ([0, T ]; Rn ) the set of n-dimensional continuous stochastic processes {ϕt ; t ∈ [0, T ]} which satisfy: 2 (iii) ϕ 2 S 2 := E (sup0≤t≤T |ϕt | ) < ∞; (iv) ϕt is Ft -measurable, for any t ∈ [0, T ]. Let
Key words. Backward doubly stochastic differential equations Comparison theorem, the multidimension, Backward stochastic integral. AMS 2000 Subject Classification: 60H10. CLC Number: O211.63
1
Introduction
Since the seminal paper on nonlinear backward stochastic differential equations (BSDEs for short) was published by Pardoux and Peng [1] in 1990, BSDEs have attracted great interest from both mathematical community and financial community (cf. [2], [3] and the references therein). In 1992, [2]Peng gave the comparison of the one-dimensional BSDE. In 1994, [4]Christel Geib, Ralf Manthey proved the comparison theorems for stochastic differential equations in finite and infinite dimensions. Applied this idea, [5]zhou got the comparison of the multi-dimensional BSDE in 1999. After they introduced the theory of BSDEs, Pardoux and Peng [6] in 1994 brought forward a new kind of BSDEs, that is a class of backward doubly stochastic differential equations (BDSDEs in short) with two different directions of stochastic integrals, i.e., the equations involve both a standard (forward) stochastic integral dWt and a backward stochastic integral dBt . They have proved the existence and uniqueness of solutions to BDSDEs under uniformly Lipschitz conditions on coefficients. That is, for a given terminal time T > 0, under the uniformly Lipschitz assumptions on coefficients f , g, for any square integrable terminal value ξ , the following BDSDE has a unique solution pair (yt , zt ) in the interval [0, T ]:
1
2
Preliminaries: the existence and uniqueness of BDSDEs and an extension of Itˆ o formula
Notation. The Euclidean norm of a vector x ∈ Rk will be denoted by |x|, and for a d × k matrix A, √ we define A = T rAA∗ , where A∗ is the transpose of A. a ∈ Rk , let ai is the number i line of a, c ∈ Rk×d, let cj is the number j line of c, ci,j is the number i line and j array of the c. For a1 , a1 ∈ Rk , we define
2 a1 ≥ a2 ⇔ a1 j ≥ aj ,
j = 1, 2, · · · .
Let (Ω, F , P ) be a probability space, and T be an arbitrarily fixed positive constant throughout this paper. Let {Wt ; 0 ≤ t ≤ T } and {Bt ; 0 ≤ t ≤ T } be two mutually independent standard Brownian Motions with values in Rd and Rl , respectively, defined on (Ω, F , P ). Let N denote the class of P -null sets of F . For each t ∈ [0, T ], we define . B Ft = FtW ∨ Ft,T
f : Ω × [0, T ] × Rk × Rk×d → Rk ,
g : Ω × [0, T ] × Rk × Rk×d → Rk×l .
be jointly measurable and satisfy the following assumption (H1) f (·, 0, 0) ∈ M 2 (0, T ; Rk ), g(·, 0, 0) ∈ M 2 (0, T ; Rk×l ).
Abstract A class of backward doubly stochastic differential equations (BDSDEs) are studied. We obtain a comparison theorem of these multi-dimension BDSDEs. We also derive the existence of solutions for this multi-dimension BDSDEs with continuous coefficients as its applications.
(H2) There exist constants C > 0 and 0 < α < 1 such that for any (ω, t) ∈ Ω × [0, T ], (y1 , z1 ), (y2 , z2 ) ∈ Rk × Rk×d , |f (t, y1 , z1 ) − f (t, y2 , z2 )|2 ≤ C (|y1 − y2 |2 + z1 − z2 2 ) g(t, y1 , z1 ) − g(t, y2 , z2 ) 2 ≤ C |y1 − y2 |2 + α z1 − z2 2 Given ξ ∈ L2 (Ω, FT , P ; Rk ), we consider the following BDSDE:
Comparison Theorems of the multi-dimensional BDSDEs and Applico Shi Yufeng Shandong Art and Design School of Mathematics and School of Mathematics and Academy Jinan 250014, China Statistics Shandong Economic System Sciences Shandong University, Jinan 250100, China University, Jinan 250100, China E-mail:byhan01@ zhubo207@ yfshi@
T T T
yt = ξ +
t
f (s, ys , zs )ds +
t
g(s, ys , zs )dBs −
t
zs dWs .
(1)
Pardoux and Peng [6] showed that BDSDEs can produce a probabilistic representation for certain quasilinear stochastic partial differential equations (SPDEs).recently Shi, Gu and Liu[7] proved the comparison theorem of these BDSDEs, but they studied the one-dimensional BDSDEs. In this paper, we applying the idea of [5]zhou, prove the comparison theorem for multi-dimensional BDSDEs. Then we also study the multi-dimension BDSDEs with continuous coefficients as an application of the comparison theorem of multi-dimension BDSDEs.
η η where for any process {ηt }, Fs,t = σ {ηr − ηs ; s ≤ r ≤ t} ∨ N , Ftη = F0 ,t . We note that the collection {Ft ; t ∈ [0, T ]} is neither increasing nor decreasing, so it does not constitute a classical filtration. For any n ∈ N , let M 2 (0, T ; Rn ) denote the set of (classes of dP ⊗ dt a.e. equal) n-dimensional jointly measurable stochastic processes {ϕt ; t ∈ [0, T ]} which satisfy: T 2 (i) ϕ 2 M 2 := E 0 |ϕt | dt < ∞; (ii) ϕt is Ft -measurable, for any t ∈ [0, T ]. Similarly, we denote by S 2 ([0, T ]; Rn ) the set of n-dimensional continuous stochastic processes {ϕt ; t ∈ [0, T ]} which satisfy: 2 (iii) ϕ 2 S 2 := E (sup0≤t≤T |ϕt | ) < ∞; (iv) ϕt is Ft -measurable, for any t ∈ [0, T ]. Let
Key words. Backward doubly stochastic differential equations Comparison theorem, the multidimension, Backward stochastic integral. AMS 2000 Subject Classification: 60H10. CLC Number: O211.63
1
Introduction
Since the seminal paper on nonlinear backward stochastic differential equations (BSDEs for short) was published by Pardoux and Peng [1] in 1990, BSDEs have attracted great interest from both mathematical community and financial community (cf. [2], [3] and the references therein). In 1992, [2]Peng gave the comparison of the one-dimensional BSDE. In 1994, [4]Christel Geib, Ralf Manthey proved the comparison theorems for stochastic differential equations in finite and infinite dimensions. Applied this idea, [5]zhou got the comparison of the multi-dimensional BSDE in 1999. After they introduced the theory of BSDEs, Pardoux and Peng [6] in 1994 brought forward a new kind of BSDEs, that is a class of backward doubly stochastic differential equations (BDSDEs in short) with two different directions of stochastic integrals, i.e., the equations involve both a standard (forward) stochastic integral dWt and a backward stochastic integral dBt . They have proved the existence and uniqueness of solutions to BDSDEs under uniformly Lipschitz conditions on coefficients. That is, for a given terminal time T > 0, under the uniformly Lipschitz assumptions on coefficients f , g, for any square integrable terminal value ξ , the following BDSDE has a unique solution pair (yt , zt ) in the interval [0, T ]:
1
2
Preliminaries: the existence and uniqueness of BDSDEs and an extension of Itˆ o formula
Notation. The Euclidean norm of a vector x ∈ Rk will be denoted by |x|, and for a d × k matrix A, √ we define A = T rAA∗ , where A∗ is the transpose of A. a ∈ Rk , let ai is the number i line of a, c ∈ Rk×d, let cj is the number j line of c, ci,j is the number i line and j array of the c. For a1 , a1 ∈ Rk , we define
2 a1 ≥ a2 ⇔ a1 j ≥ aj ,
j = 1, 2, · · · .
Let (Ω, F , P ) be a probability space, and T be an arbitrarily fixed positive constant throughout this paper. Let {Wt ; 0 ≤ t ≤ T } and {Bt ; 0 ≤ t ≤ T } be two mutually independent standard Brownian Motions with values in Rd and Rl , respectively, defined on (Ω, F , P ). Let N denote the class of P -null sets of F . For each t ∈ [0, T ], we define . B Ft = FtW ∨ Ft,T
f : Ω × [0, T ] × Rk × Rk×d → Rk ,
g : Ω × [0, T ] × Rk × Rk×d → Rk×l .
be jointly measurable and satisfy the following assumption (H1) f (·, 0, 0) ∈ M 2 (0, T ; Rk ), g(·, 0, 0) ∈ M 2 (0, T ; Rk×l ).
Abstract A class of backward doubly stochastic differential equations (BDSDEs) are studied. We obtain a comparison theorem of these multi-dimension BDSDEs. We also derive the existence of solutions for this multi-dimension BDSDEs with continuous coefficients as its applications.