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 Electronic Journal of Probability > Vol. 13 (2008) > Paper 35 open journal systems 


Sobolev solution for semilinear PDE with obstacle under monotonicity condition

Anis Matoussi, Université du Maine
Mingyu Xu, Institute of Applied Mathematics, Beijing


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


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Pages: 1035-1067

Published on: June 29, 2008
Bibliography
  1. Bally V., Caballero E., El-Karoui N. and B.Fernandez : Reflected BSDE's PDE's and Variational Inequalities (to appear in Bernoulli 2007).
  2. 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. MR1752675 (2000m:60067)
  3. Bally, V.; Matoussi, A. Weak solutions for SPDEs and backward doubly stochastic differential equations. J. Theoret. Probab. 14 (2001), no. 1, 125--164. MR1822898 (2001m:60142)
  4. 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 MR0513618 (58 #23923)
  5. Bismut, Jean-Michel. Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44 (1973), 384--404. MR0329726 (48 #8067)
  6. El Karoui, N.; Peng, S.; Quenez, M. C. Backward stochastic differential equations in finance. Math. Finance 7 (1997), no. 1, 1--71. MR1434407 (98d:90030)
  7. 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. MR1434123 (98k:60096)
  8. 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. MR1470516 (99b:60078)
  9. Hamadène, S.; Ouknine, Y. Reflected backward stochastic differential equation with jumps and random obstacle. Electron. J. Probab. 8 (2003), no. 2, 20 pp. (electronic). MR1961164 (2003m:60148)
  10. 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).
  11. 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. MR0876080 (87m:60127)
  12. 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. MR2135157 (2006c:60070)
  13. 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 (2000e:60096)
  14. Pardoux, É.; Peng, S. G. Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 (1990), no. 1, 55--61. MR1037747 (91e:60171)
  15. 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. MR1176785 (93k:60157)
  16. Peng, Shi Ge. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics Stochastics Rep. 37 (1991), no. 1-2, 61--74. MR1149116 (93a:35159)
  17. Revuz D. and Yor, M. : Continuous martingales and Brownian motion (Springer, Berlin) (1991).
  18. 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 MR1696772 (2001d:93004)













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Electronic Journal of Probability. ISSN: 1083-6489