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The obstacle problem for quasilinear stochastic PDE’s. (English) Zbl 1200.60052
Given a deterministic terminal condition \(\Phi\) at a time \(T>0\) and an obstacle \(v\), existence of a unique solution \((u,\nu)\) of an equation \[ du+[\frac12\Delta u+f(t,x,u,\nabla u)+\text{div}g(t,x,u,\nabla u)]\,dt+h(t,x,u,\nabla u)\cdot\overleftarrow{dB}=-\nu(dt\,dx) \] with the constraint \(\nu(u>v)=0\) is established.
The random functions \(f\), \(g\), \(h\) are assumed to be Lipschitz (with sufficiently small Lipschitz constants), \(B\) is a finite dimensional Wiener process, \(\int X\cdot\overleftarrow{dB}\) denotes the backward stochastic integral, the obstacle \(v=v(\omega,t,x)\) is a predictable function (e.g. continuous in \((t,x)\)) satisfying \(\Phi\geq v\).
The above equation is understood in the weak PDE sense (using space-time test functions) and a solution has two components where \(u\) is a process in the Sobolev space \(H^1\) with \(L^2\)-continuous paths, \(\nu\) is a non-negative random measure on \([0,T]\times\mathbb R^d\) and \(u\) and \(\nu\) are linked by a certain joint regularity condition (expressed in terms of the stochastic potential theory).
The paper is presented in a detailed way and the proof is based on a penalization method and the theory of potential.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G46 Martingales and classical analysis
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