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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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##### References:
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