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Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. (English) Zbl 0899.60047
Authors’ abstract: We study reflected solutions of one-dimensional stochastic differential equations. The “reflection” keeps the solution above a given stochastic process. We prove uniqueness and existence both by a fixed point argument and by approximation via penalization. We show that when the coefficient has a special form, then the solution of our problem is the value function of a mixed optimal stopping-optimal stochastic control problem. We finally show that, when put in a Markovian framework, the solution of our BSDE provides a probabilistic formula for the unique viscosity solution of an obstacle problem for a parabolic partial differential equation.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
60H30 Applications of stochastic analysis (to PDEs, etc.)
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[1] Barles, G. (1994). Solutions de Viscosité des Équations de Hamilton-Jacobi du Premier Ordre et Applications. Springer, New York.
[2] Barles, G. and Burdeau, J. (1995). The Dirichlet problem for semilinear second order degenerate elliptic equations and applications to stochastic exit time problems. Comm. Partial Differ. Equations 20 129-178. · Zbl 0826.35038
[3] Bensoussan, A. and Lions, J. L. (1978). Applications des Inéquations Varitionelles en Contr ole Stochastique. Dunod, Paris. · Zbl 0411.49002
[4] Crandall, M., Ishii, H. and Lions, P. L. (1992). User’s guide to the viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 1-67. · Zbl 0755.35015
[5] Darling, R. (1995). Constructing gamma martingales with prescribed limits, using backward SDEs. Ann. Probab. 23 1234-1261. · Zbl 0839.58060
[6] Davis, M. and Karatzas, I. (1994). A deterministic approach to optimal stopping. In Probability, Statistics and Optimization (F. P. Kelly, ed.) 455-466. Wiley, New York. · Zbl 0855.60041
[7] Dellacherie, C. and Meyer, P. A. (1975). Probabilités et Potentiel. I-IV. Hermann, Paris. · Zbl 0323.60039
[8] Dellacherie, C. and Meyer, P. A. (1980). Probabilités et Potentiel. V-VIII. Hermann, Paris. · Zbl 0464.60001
[9] Duffie, D. and Epstein, L. (1992). Stochastic differential utility. Econometrica 60 353-394. · Zbl 0768.90006
[10] El Karoui, N. (1981). Les aspects probabilistes du contr ole stochastique. In Ecole d’Eté de Saint Flour 1979. Lecture Notes in Math. 876. Springer, Berlin. · Zbl 0472.60002
[11] El Karoui, N. and Chaleyat-Maurel, M. (1978). Un probl eme de réflexion et ses applications au temps local et aux equations differentielles stochastiques sur R. Cas continu. In Temps Locaux. Astérisque 52-53 117-144. Soc. Math. France, Paris.
[12] El Karoui, N. and Jeanblanc-Picqué, M. (1993). Optimization of consumption with labor income. Unpublished manuscript. · Zbl 0930.60050
[13] El Karoui, N., Peng, S. and Quenez, M. C. (1994). Backward stochastic differential equations in finance. Math. Finance. · Zbl 0884.90035
[14] Hamadene, S. and Lepeltier, J. P. (1995). Zero-sum stochastic differential games and backward equations. Systems Control Lett. 24 259-263. · Zbl 0877.93125
[15] Jacka, S. (1993). Local times, optimal stopping and semimartingales. Ann. Probab. 21 329- 339. · Zbl 0773.60031
[16] Karatzas, I. and Shreve, S. (1996). Mathematical Finance. · Zbl 0941.91032
[17] Pardoux, E. and Peng, S. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55-61. · Zbl 0692.93064
[18] Pardoux, E. and Peng, S. (1992). Backward SDEs and quasilinear PDEs. In Stochastic Partial Differential Equations and Their Applications (B. L. Rozovskii and R. B. Sowers, eds.). Lecture Notes in Control and Inform. Sci. 176. Springer, Berlin. · Zbl 0766.60079
[19] Pardoux, E. and Peng, S. (1996). Some backward SDEs with non-Lipschitz coefficients. Proc. Conf. Metz.
[20] Revuz, D. and Yor, M. (1994). Continuous Martingales and Brownian Motion. Springer, New York. · Zbl 0804.60001
[21] LATP, URA CNRS 225 Centre de Mathématiques et d’Informatique Université de Provence 39, rue F. JoliotCurie F13453 Marseille cedex 13 France E-mail: pardoux@gyptis.univ-mrs.fr S. Peng Institute of Mathematics Shandong University Jinan, 250100 China E-mail: pengsg@shandong.ihep.ac.cn
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