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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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