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Reflected backward doubly stochastic differential equations with discontinuous generator. (English) Zbl 1284.60113
The authors consider the reflected backward doubly stochastic differential equations (RBDSDEs):
(i) \(Y_t= \xi+\int^T_t f(s, Y_s, Z_s)\,ds+ \int^T_t g(s, Y_s, Z_s)\,d\overline B_s+ K_t- K_t- \int^T_t Z_s\,dW_s\),
(ii) \(Y_t\geq S_t\),
(iii) \(K\) is a nondecreasing process such that \(K_0= 0\) and \(\int^T_t (Y_t- S_t)\,dK_t= 0\),
where \(\int^T_t Z_s\,dW_s\) is a standard forward stochastic Itō integral and \(\int^T_t g(s, Y_s, Z_s)\,d\overline B_s\) is a backward stochastic Kunita-Itō integral.
Let \(M^2(0, T,\mathbb{R}^n)\) denote the class of \(n\)-dimensional jointly measurable random processes \(\{\varphi_t:\,0\leq t\leq T\}\) which satisfy:
\(\|\varphi\|^2_{M^2}= \operatorname{E}(\int^T_0 |\varphi_t|^2\,dt)<\infty\),
\(\varphi_t\) is \(F_t\)-measurable for a.e. \(t\in [0,T]\),
let \(S^2([0,T],\mathbb{R}^n)\) denote the set of continuous \(n\)-dimensional random processes which satisfy:
\(\|\varphi\|^2_{S^2}= \operatorname{E}(\sup_{0\leq t\leq T}|\varphi_t|^2)<\infty\),
\(\varphi_t\) is \(F_t\)-measurable for a.e. \(t\in [0,T]\),
let \(A^2(0,T)\) denote the space of a continuous, real-valued, increasing processes such that \(\varphi_0= 0\) and
\(|\|\varphi\|^2_{A^2}= \operatorname{E}(|\varphi_T|^2)<\infty\),
\(\varphi_t\) is \(F_t\)-measurable for a.e. \(t\in [0,T]\).
A solution of a RBDSDE is a triple of processes \((Y,Z,K)\) satisfying (i), (ii), (iii) in the space \(S^2([0,T], \mathbb{R}^n)\times M^2(0, T,\mathbb{R})\times A^2(0,T)\).
The triplet of processes \((Y_*, Z_*, K_*)\) is said to be a minimal solution of RBDSDE (i), (ii), (iii) if for other any solution \((Y,Z,K)\), we have \(Y_*\leq Y\).
Under certain assumptions on the random variable \(\xi\), functions \(f\) and \(g\) and a process \(S\), the authors show that problem (i), (ii), (iii) has a minimal solution \((Y_*, Z_*, K_*)\).

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H20 Stochastic integral equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
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