Deterministic and stochastic differential inclusions with multiple surfaces of discontinuity
The authors study a constrained discontinuous media problem (CDMP), that includes differential inclusions and stochastic differential equations with discontinuous drift \(f\) and solutions are constrained to live in a given closed domain \(G\) in \(\mathbb{R}^n\) according to a constrain vector field \(D(\cdot)\) specified on the boundary \(\partial D\) of the domain. More exactly, it is considered a equation by the following form: \[ \phi=\theta+\eta+u,\quad \dot{\theta}(t)\in F(\phi(t)),\quad \text{a.e. }t \] for \(u\) in an appropriate class of functions, where \(\eta\) in the constraining term in the Skorokhod problem specified by \((G,D)\) and \(F\) is the set-valued upper semicontinuous envelope of \(f\). The paper treats the case when \(f\) has non-trivial state-dependence and thus extend other works in this framework, and yields new existence and uniqueness results. One of the main results is Theorem 2.3, which establishes existence of the solution to the CDMP under two natural assumptions on the constrained data \((G,D)\) and under a mild condition of \(F\). Next, it is proved an uniqueness result for the class of polyhedral CDMPs with two additional assumptions (analogues with there of [\textit{P. Dupuis} and \textit{H. Ishii}, Stochastics Stochastics Rep. 35, No. 1, 31--62 (1991; Zbl 0721.60062), Assumption 2.1]). Five very illustrative examples show the consistency of the given results. The another main result presents a very important study on some strong existence and pathwise uniqueness for a family of stochastic differential equations with reflection and a discontinuous drift (Theorem 3.4) under some Lipschitz conditions of the function \(\phi\) and the diffusion coefficient function \(\sigma\). The presented result extend some other works [see \textit{A. Yu. Veretennikov} [Math. USSR, Sb. 39, 387--403 (1981), translation from Mat. Sb., Nov. Ser. 111(153), 434--452 (1980; Zbl 0431.60061); Izv. Akad. Nauk SSSR, Ser. Mat. 47, No. 1, 189--196 (1983; Zbl 0514.60056)]or \textit{E. Cépa} [Ann. Probab. 26, 500--532 (1998; Zbl 0937.34046)].
scientific article; zbMATH DE number 5351828
scientific article