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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.

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G46 Martingales and classical analysis
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[1] Bally, V., Caballero, E., El Karoui, N. and Fernandez, B. (2004). Reflected BSDE’s PDE’s and variational inequalities. Report, INRIA.
[2] Bally, V. and Matoussi, A. (2001). Weak solutions for SPDEs and backward doubly stochastic differential equations. J. Theoret. Probab. 14 125-164. · Zbl 0982.60057 · doi:10.1023/A:1007825232513
[3] Bensoussan, A. and Lions, J.-L. (1978). Applications des Inéquations Variationnelles en Contrôle Stochastique. Méthodes Mathématiques de l’Informatique 6 . Dunod, Paris. · Zbl 0411.49002
[4] Blumenthal, R. M. and Getoor, R. K. (1968). Markov Processes and Potential Theory. Pure and Applied Mathematics 29 . Academic Press, New York. · Zbl 0169.49204 · www.sciencedirect.com
[5] Brézis, H. (2005). Analyse Fonctionnelle-Theorie et Application . Dunod, Paris.
[6] Dellacherie, C. and Meyer, P.-A. (1975). Probabilités et Potentiel. Chapitres XII à XVI. Actualités Scientifiques et Industrielles 1372 . Hermann, Paris. · Zbl 0323.60039
[7] Denis, L. and Stoica, L. (2004). A general analytical result for non-linear SPDE’s and applications. Electron. J. Probab. 9 674-709 (electronic). · Zbl 1067.60048 · emis:journals/EJP-ECP/_ejpecp/EjpVol9/paper23.abs.html · eudml:124919
[8] Donati-Martin, C. and Pardoux, É. (1993). White noise driven SPDEs with reflection. Probab. Theory Related Fields 95 1-24. · Zbl 0794.60059 · doi:10.1007/BF01197335
[9] El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. and Quenez, M. C. (1997). Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. Ann. Probab. 25 702-737. · Zbl 0899.60047 · doi:10.1214/aop/1024404416
[10] Fukushima, M., Ōshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes. de Gruyter Studies in Mathematics 19 . de Gruyter, Berlin. · Zbl 0838.31001
[11] Matoussi, A. and Scheutzow, M. (2002). Stochastic PDEs driven by nonlinear noise and backward doubly SDEs. J. Theoret. Probab. 15 1-39. · Zbl 0999.60063 · doi:10.1023/A:1013803104760
[12] Matoussi, A. and Xu, M. (2008). Sobolev solution for semilinear PDE with obstacle under monotonicity condition. Electron. J. Probab. 13 1035-1067. · Zbl 1191.35133 · emis:journals/EJP-ECP/_ejpecp/viewarticle97da.html · eudml:233051
[13] Matoussi, A. and Xu, M. (2010). Reflected backward doubly SDE and obstacle problem for semilinear stochastic PDE’s. Preprint, Univ. Maine.
[14] Mignot, F. and Puel, J.-P. (1975). Solution maximum de certaines inéquations d’évolution paraboliques, et inéquations quasi variationnelles paraboliques. C. R. Acad. Sci. Paris Sér. A-B 280 A259-A262. · Zbl 0305.35060
[15] Nualart, D. and Pardoux, É. (1992). White noise driven quasilinear SPDEs with reflection. Probab. Theory Related Fields 93 77-89. · Zbl 0767.60055 · doi:10.1007/BF01195389
[16] Pardoux, É. and Peng, S. G. (1994). Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probab. Theory Related Fields 98 209-227. · Zbl 0792.60050 · doi:10.1007/BF01192514
[17] Stoica, I. L. (2003). A probabilistic interpretation of the divergence and BSDE’s. Stochastic Process. Appl. 103 31-55. · Zbl 1075.60544 · doi:10.1016/S0304-4149(02)00179-5
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