Electronic Journal of Probability

Sobolev solution for semilinear PDE with obstacle under monotonicity condition

Anis Matoussi and Mingyu Xu

Full-text: Open access

Abstract

We prove the existence and uniqueness of Sobolev solution of a semilinear PDE's and PDE's with obstacle under monotonicity condition. Moreover we give the probabilistic interpretation of the solutions in term of Backward SDE and reflected Backward SDE respectively

Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 35, 1035-1067.

Dates
Accepted: 29 June 2008
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819108

Digital Object Identifier
doi:10.1214/EJP.v13-522

Mathematical Reviews number (MathSciNet)
MR2424986

Zentralblatt MATH identifier
1191.35133

Subjects
Primary: 35D05
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30B

Keywords
Backward stochastic differential equation Reflected backward stochastic differential equation monotonicity condition Stochastic flow partial differential equation with obstacle

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Matoussi, Anis; Xu, Mingyu. Sobolev solution for semilinear PDE with obstacle under monotonicity condition. Electron. J. Probab. 13 (2008), paper no. 35, 1035--1067. doi:10.1214/EJP.v13-522. https://projecteuclid.org/euclid.ejp/1464819108


Export citation

References

  • Bally V., Caballero E., El-Karoui N. and B.Fernandez : Reflected BSDE's PDE's and Variational Inequalities (to appear in Bernoulli 2007).
  • Barles, G.; Lesigne, E. SDE, BSDE and PDE. Backward stochastic differential equations (Paris, 1995–1996), 47–80, Pitman Res. Notes Math. Ser., 364, Longman, Harlow, 1997.
  • Bally, V.; Matoussi, A. Weak solutions for SPDEs and backward doubly stochastic differential equations. J. Theoret. Probab. 14 (2001), no. 1, 125–164.
  • Bensoussan, A.; Lions, J.-L. Applications des inéquations variationnelles en contrôle stochastique.(French) Méthodes Mathématiques de l'Informatique, No. 6.Dunod, Paris, 1978. viii+545 pp. ISBN: 2-04-010336-8
  • Bismut, Jean-Michel. Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44 (1973), 384–404.
  • El Karoui, N.; Peng, S.; Quenez, M. C. Backward stochastic differential equations in finance. Math. Finance 7 (1997), no. 1, 1–71.
  • 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.
  • El Karoui, N.; Pardoux, E.; Quenez, M. C. Reflected backward SDEs and American options. Numerical methods in finance, 215–231, Publ. Newton Inst., Cambridge Univ. Press, Cambridge, 1997.
  • Hamadène, S.; Ouknine, Y. Reflected backward stochastic differential equation with jumps and random obstacle. Electron. J. Probab. 8 (2003), no. 2, 20 pp. (electronic).
  • Hamadène S., Lepeltier, J.-P. and Matoussi A. : Doublebarriers reflected backward SDE's with continuous coefficients. Pitman Research Notes in Mathematics Series,364, 115-128 (1997).
  • Kunita, H. Stochastic differential equations and stochastic flows of diffeomorphisms. École d'été de probabilités de Saint-Flour, XII–-1982, 143–303, Lecture Notes in Math., 1097, Springer, Berlin, 1984.
  • Lepeltier, J.-P.; Matoussi, A.; Xu, M. Reflected backward stochastic differential equations under monotonicity and general increasing growth conditions. Adv. in Appl. Probab. 37 (2005), no. 1, 134–159.
  • 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.
  • Pardoux, É.; Peng, S. G. Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 (1990), no. 1, 55–61.
  • Pardoux, É.; Peng, S. Backward stochastic differential equations and quasilinear parabolic partial differential equations. Stochastic partial differential equations and their applications (Charlotte, NC, 1991), 200–217, Lecture Notes in Control and Inform. Sci., 176, Springer, Berlin, 1992.
  • Peng, Shi Ge. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics Stochastics Rep. 37 (1991), no. 1-2, 61–74.
  • Revuz D. and Yor, M. : Continuous martingales and Brownian motion (Springer, Berlin) (1991).
  • Yong, Jiongmin; Zhou, Xun Yu. Stochastic controls.Hamiltonian systems and HJB equations.Applications of Mathematics (New York), 43. Springer-Verlag, New York, 1999. xxii+438 pp. ISBN: 0-387-98723-1