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The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations. (English) Zbl 0744.60065

The Skorokhod oblique reflection problem for \((D,\Gamma,w)\) (\(D\) a general domain in \(\mathbb{R}^ d\), \(\Gamma(x)\), \(x\in\partial D\), a convex cone of directions of reflection, \(w\) a function in \({\mathcal D}(\mathbb{R}^ +,\mathbb{R}^ d)\)) is considered. It is first proved, under a condition on \((D,\Gamma)\), corresponding to \(\Gamma(x)\) not being simultaneously too large and too much skewed with respect to \(\partial D\), that given a sequence \(\{w^ n\}\) of functions converging in the Skorokhod topology to \(w\), any sequence \(\{(x^ n,\varphi^ n)\}\) of solutions to the Skorokhod problem for \((D,\Gamma,w^ n)\) is relatively compact and any of its limit points is a solution to the Skorokhod problem for \((D,\Gamma,w)\). Next it is shown that if \((D,\Gamma)\) satisfies the uniform exterior sphere condition and another requirement, then solutions to the Skorokhod problem for \((D,\Gamma,w)\) exist for every \(w\in{\mathcal D}(\mathbb{R}^ +,\mathbb{R}^ d)\) with small enough jump size. The requirement is met in the case when \(\partial D\) is piecewise \({\mathcal C}^ 1_ b\), \(\Gamma\) is generated by continuous vector fields on the faces of \(D\) and \(\Gamma(x)\) makes an angle (in in a suitable sense) of less than \(\pi/2\) with the cone of inward normals at \(D\), for every \(x\in\partial D\). Existence of obliquely reflecting Brownian motion and of weak solutions to stochastic differential equations with oblique reflection boundary conditions is derived.

MSC:

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