Penalization method for a nonlinear Neumann PDE via weak solutions of reflected SDEs

Khaled Bahlali (Université de Toulon (IMATH) and Aix-Marseille Université (CNRS, LATP))
Lucian Maticiuc ("Alexandru Ioan Cuza" University of Iasi and “Gheorghe Asachi” Technical University)
Adrian Zalinescu ("Alexandru Ioan Cuza" University of Iasi and “Octav Mayer” Mathematics Institute of the Romanian Academy)

Abstract


In this paper we prove an approximation result for the viscosity solution of a system of semi-linear partial differential equations with continuous coefficients and nonlinear Neumann boundary condition. The approximation we use is based on a penalization method and our approach is probabilistic. We prove the weak uniqueness of the solution for the reflected stochastic differential equation and we approximate it (in law) by a sequence of solutions of stochastic differential equations with penalized terms. Using then a suitable generalized backward stochastic differential equation and the uniqueness of the reflected stochastic differential equation, we prove the existence of a continuous function, given by a probabilistic representation, which is a viscosity solution of the considered partial differential equation. In addition, this solution is approximated by the penalized partial differential equation.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-19

Publication Date: November 27, 2013

DOI: 10.1214/EJP.v18-2467

References

  • Alibert, Jean-Jacques; Bahlali, Khaled. Genericity in deterministic and stochastic differential equations. Séminaire de Probabilités, XXXV, 220--240, Lecture Notes in Math., 1755, Springer, Berlin, 2001. MR1837290
  • Bahlali, Khaled; Elouaflin, Abouo; Pardoux, Étienne. Homogenization of semilinear PDEs with discontinuous averaged coefficients. Electron. J. Probab. 14 (2009), no. 18, 477--499. MR2480550
  • Boufoussi, Brahim; Van Casteren, Jan. An approximation result for a nonlinear Neumann boundary value problem via BSDEs. Stochastic Process. Appl. 114 (2004), no. 2, 331--350. MR2101248
  • El Karoui, Nicole. Backward stochastic differential equations: a general introduction. Backward stochastic differential equations (Paris, 1995–1996), 7--26, Pitman Res. Notes Math. Ser., 364, Longman, Harlow, 1997. MR1752672
  • El Karoui, Nicole. Processus de reflexion dans $R^{n}$. (French) Séminaire de Probabilités, IX (Seconde Partie, Univ. Strasbourg, Strasbourg, années universitaires 1973/1974 et 1974/1975), pp. 534--554. Lecture Notes in Math., Vol. 465, Springer, Berlin, 1975. MR0423554
  • Jakubowski, Adam. A non-Skorohod topology on the Skorohod space. Electron. J. Probab. 2 (1997), no. 4, 21 pp. (electronic). MR1475862
  • Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1988. xxiv+470 pp. ISBN: 0-387-96535-1 MR0917065
  • Łaukajtys, Weronika; Słomiński, Leszek. Penalization methods for reflecting stochastic differential equations with jumps. Stoch. Stoch. Rep. 75 (2003), no. 5, 275--293. MR2017780
  • Łaukajtys, Weronika; Słomiński, Leszek. Penalization methods for the Skorokhod problem and reflecting SDEs with jumps. Bernoulli 19 (2013), no. 5A, 1750-1775.
  • Lejay, Antoine. BSDE driven by Dirichlet process and semi-linear parabolic PDE. Application to homogenization. Stochastic Process. Appl. 97 (2002), no. 1, 1--39. MR1870718
  • Lions, Pierre-Louis; Menaldi, José-Luis; Sznitman, Alain-Sol. Construction de processus de diffusion réfléchis par pénalisation du domaine. (French) C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 11, 559--562. MR0614669
  • Lions, Pierre-Louis; Sznitman, Alain-Sol. Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 (1984), no. 4, 511--537. MR0745330
  • Maticiuc, Lucian; Răşcanu, Aurel. Viability of moving sets for a nonlinear Neumann problem. Nonlinear Anal. 66 (2007), no. 7, 1587--1599. MR2301340
  • Menaldi, José-Luis. Stochastic variational inequality for reflected diffusion. Indiana Univ. Math. J. 32 (1983), no. 5, 733--744. MR0711864
  • Pardoux, Étienne. BSDEs, weak convergence and homogenization of semilinear PDEs. Nonlinear analysis, differential equations and control (Montreal, QC, 1998), 503--549, NATO Sci. Ser. C Math. Phys. Sci., 528, Kluwer Acad. Publ., Dordrecht, 1999. MR1695013
  • Pardoux, Étienne; Zhang, Shuguang. Generalized BSDEs and nonlinear Neumann boundary value problems. Probab. Theory Related Fields 110 (1998), no. 4, 535--558. MR1626963
  • Pardoux, Étienne; Răşcanu, Aurel. Stochastic differential equations, Backward SDEs, Partial differential equations. Stochastic Modelling and Applied Probability, Springer, in press, 2014.
  • Rozkosz, Andrzej; Słomiński, Leszek. On stability and existence of solutions of SDEs with reflection at the boundary. Stochastic Process. Appl. 68 (1997), no. 2, 285--302. MR1454837
  • Skorokhod, A. V. Studies in the theory of random processes. Translated from the Russian by Scripta Technica, Inc. Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965 viii+199 pp. MR0185620
  • Stricker, Christophe. Lois de semimartingales et critères de compacité. (French) [Laws of semimartingales and compactness criteria] Séminaire de probabilités, XIX, 1983/84, 209--217, Lecture Notes in Math., 1123, Springer, Berlin, 1985. MR0889478
  • Stroock, Daniel W.; Varadhan, S. R. S. Diffusion processes with boundary conditions. Comm. Pure Appl. Math. 24 1971 147--225. MR0277037
  • Tanaka, Hiroshi. Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9 (1979), no. 1, 163--177. MR0529332
  • Zălinescu, Adrian. Weak solutions and optimal control for multivalued stochastic differential equations. NoDEA Nonlinear Differential Equations Appl. 15 (2008), no. 4-5, 511--533. MR2465976
Bookmark and Share


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