Electronic Journal of Probability

Maximum principle for quasilinear stochastic PDEs with obstacle

Laurent Denis, Anis Matoussi, and Jing Zhang

Full-text: Open access


We prove a maximum principle for local solutions of quasi linear stochastic PDEs with obstacle (in short OSPDE). The proofs are based on a version of Ito's formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 44, 32 pp.

Accepted: 12 May 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 31B150

Stochastic PDE's Obstacle problems It\^o's formula $L^p-$estimate Local solution Comparison theorem Maximum principle Moser iteration

This work is licensed under a Creative Commons Attribution 3.0 License.


Denis, Laurent; Matoussi, Anis; Zhang, Jing. Maximum principle for quasilinear stochastic PDEs with obstacle. Electron. J. Probab. 19 (2014), paper no. 44, 32 pp. doi:10.1214/EJP.v19-2716. https://projecteuclid.org/euclid.ejp/1465065686

Export citation


  • Aronson, D. G. Non-negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 607–694.
  • Aronson, D. G.; Serrin, James. Local behavior of solutions of quasilinear parabolic equations. Arch. Rational Mech. Anal. 25 1967 81–122.
  • Bally V., Caballero E., El-Karoui N. and Fernandez, B. : Reflected BSDE's PDE's and Variational Inequalities. textitpreprint INRIA report (2004). x3-827 (2004).
  • Denis, Laurent; Stoica, L. A general analytical result for non-linear SPDE's and applications. Electron. J. Probab. 9 (2004), no. 23, 674–709 (electronic).
  • Denis, Laurent; Matoussi, Anis; Stoica, Lucretiu. $L^ p$ estimates for the uniform norm of solutions of quasilinear SPDE's. Probab. Theory Related Fields 133 (2005), no. 4, 437–463.
  • Denis L., Matoussi A. and Stöica L.: Maximum principle for parabolic SPDE's: first approach. textitStohcastic Partial Differential Equations and Applications VIII, Levico, Jan. 6-12 (2008).
  • Denis, Laurent; Matoussi, Anis; Stoica, Lucretiu. Maximum principle and comparison theorem for quasi-linear stochastic PDE's. Electron. J. Probab. 14 (2009), no. 19, 500–530.
  • Denis, Laurent; Matoussi, Anis. Maximum principle for quasilinear SPDE's on a bounded domain without regularity assumptions. Stochastic Process. Appl. 123 (2013), no. 3, 1104–1137.
  • Denis L., Matoussi A. and Zhang J.: The Obstacle Problem for Quasilinear Stochastic PDEs: Analytical approach. To appear in Annals of Probability (2013).
  • Donati-Martin, C.; Pardoux, É. White noise driven SPDEs with reflection. Probab. Theory Related Fields 95 (1993), no. 1, 1–24.
  • El Karoui, N.; Kapoudjian, C.; Pardoux, E.; Peng, S.; Quenez, M. C. Reflected solutions of backward SDE's, and related obstacle problems for PDE's. Ann. Probab. 25 (1997), no. 2, 702–737.
  • Klimsiak, Tomasz. Reflected BSDEs and the obstacle problem for semilinear PDEs in divergence form. Stochastic Process. Appl. 122 (2012), no. 1, 134–169.
  • Krylov, N. V. An analytic approach to SPDEs. Stochastic partial differential equations: six perspectives, 185–242, Math. Surveys Monogr., 64, Amer. Math. Soc., Providence, RI, 1999.
  • Lions J.L. and Magenes E.: Problémes aux limites non homogènes et applications. Dunod, Paris (1968).
  • Matoussi, Anis; Stoica, Lucretiu. The obstacle problem for quasilinear stochastic PDE's. Ann. Probab. 38 (2010), no. 3, 1143–1179.
  • Mignot, F.; Puel, J.-P. Inéquations d'évolution paraboliques avec convexes dépendant du temps. Applications aux inéquations quasi variationnelles d'évolution. (French) Arch. Rational Mech. Anal. 64 (1977), no. 1, 59–91.
  • Nualart, D.; Pardoux, É. White noise driven quasilinear SPDEs with reflection. Probab. Theory Related Fields 93 (1992), no. 1, 77–89.
  • Pierre, Michel. Problémes d'evolution avec contraintes unilatérales et potentiels paraboliques. (French) Comm. Partial Differential Equations 4 (1979), no. 10, 1149–1197.
  • Pierre, Michel. Représentant précis d'un potentiel parabolique. (French) Seminar on Potential Theory, Paris, No. 5 (French), pp. 186–228, Lecture Notes in Math., 814, Springer, Berlin, 1980.
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7
  • Sanz-Solé, Marta; Vuillermot, Pierre-A. Equivalence and Hölder-Sobolev regularity of solutions for a class of non-autonomous stochastic partial differential equations. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), no. 4, 703–742.
  • Xu, Tiange; Zhang, Tusheng. White noise driven SPDEs with reflection: existence, uniqueness and large deviation principles. Stochastic Process. Appl. 119 (2009), no. 10, 3453–3470.