One barrier reflected backward doubly stochastic differential equations with discontinuous monotone coefficients.

*(English)*Zbl 1261.60056This paper studies reflected doubly stochastic backward differential equations of the form
\[
y_t=\xi + \int_{t}^{T}f(s,y_s,z_s)\,ds+ \int_{t}^{T}g(s,y_s,z_s)dB_s-\int_{t}^{T}z_s dW_s +K_T-K_t,
\]
\[
y_t \geq S_t,\;K_0=0,\;\int_{0}^{T}(y_t-S_t)dK_t =0,
\]
where the first stochastic integral is a backward Itō integral and the second one is a standard forward Itō integral. In the above \(S_t\) is given and the equation has to be solved for the triple \((y_t,z_t,K_t)\). \(S_t\) is a given stochastic process such that the unknown process \(y_t\) must always stay above it (called the obstacle) whereas \(K_t\) is a reflecting process. It is the aim of the paper to establish existence and comparison results for the solutions of this equation under the assumption of discontinuous and monotone coefficients.

The proof employs approximation techniques. For example, the generator \(f\) is approximated by an increasing sequence of Lipschitz functions satisfying certain boundedness properties. A sequence of approximation problems is then formulated the solution of which enjoys certain monotonicity and comparison properties that allow us to go to the limit in a meaningful way and to obtain the solution of the original problem. The comparison results proceed in a similar fashion.

The proof employs approximation techniques. For example, the generator \(f\) is approximated by an increasing sequence of Lipschitz functions satisfying certain boundedness properties. A sequence of approximation problems is then formulated the solution of which enjoys certain monotonicity and comparison properties that allow us to go to the limit in a meaningful way and to obtain the solution of the original problem. The comparison results proceed in a similar fashion.

Reviewer: Athanasios Yannacopoulos (Athens)

##### MSC:

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

##### Keywords:

reflected backward doubly stochastic differential equations; monotone coefficients; comparison results
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\textit{Z. Li} and \textit{J. Luo}, Stat. Probab. Lett. 82, No. 10, 1841--1848 (2012; Zbl 1261.60056)

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