One barrier reflected backward doubly stochastic differential equations with discontinuous monotone coefficients
reflected backward doubly stochastic differential equations
This paper studies reflected doubly stochastic backward differential equations of the form NEWLINE\[NEWLINEy_t=\xi + \int_{t}^{T}f(s,y_s,z_s)\,ds+ \int_{t}^{T}g(s,y_s,z_s)dB_s-\int_{t}^{T}z_s dW_s +K_T-K_t,NEWLINE\]NEWLINE NEWLINE\[NEWLINE y_t \geq S_t,\;K_0=0,\;\int_{0}^{T}(y_t-S_t)dK_t =0,NEWLINE\]NEWLINE 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.NEWLINENEWLINEThe 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.
scientific article; zbMATH DE number 6085261
scientific article