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Backward doubly stochastic differential equations with weak assumptions on the coefficients. (English) Zbl 1220.60035

The author considers the following one-dimensional nonlinear backward double stochastic differential equations (BDSDE) \[ y_t= y_0+ \int^T_t (\text{sgn}(y_s) y^2_s+ \sqrt{z_s\mathbf{1}_{\{z_s\geq 0\}}})\,ds+ \int^T_t g(s, y_s,z_s)\,dB_s+ \int^T_t z_s dw_s,\;t\in [0,T], \] where \(g: [0, T]\times\mathbb{R}\times \mathbb{R}^w\to \mathbb{R}^B\) is a nonlinear vector function, \(\{w_t\}){0\leq t\leq T}\) and \(\{w_t\}_{0\leq t\leq T}\) are two mutually independent Wiener processes with values in \(\mathbb{R}^w\) and \(\mathbb{R}^B\), respectively; the integrals with respect to \(\{B_t\}\) and \(\{B_t\}\) is a backward Itô’s integral and standard forward Itô’s integral, respectively.
The author obtain as well existence theorems for BDSDEs with general continuous coefficients as an existence theorem and comparisons theorems for BDSDEs with discontinuous coefficients. Some remarks (examples) illustrate the obtained results.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H05 Stochastic integrals
60H99 Stochastic analysis
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