The Annals of Probability

Reflected solutions of backward SDE's, and related obstacle problems for PDE's

N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng, and M. C. Quenez

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Abstract

We study reflected solutions of one-dimensional backward 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 reflected BSDE provides a probabilistic formula for the unique viscosity solution of an obstacle problem for a parabolic partial differential equation.

Article information

Source
Ann. Probab., Volume 25, Number 2 (1997), 702-737.

Dates
First available in Project Euclid: 18 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1024404416

Digital Object Identifier
doi:10.1214/aop/1024404416

Mathematical Reviews number (MathSciNet)
MR1434123

Zentralblatt MATH identifier
0899.60047

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.) 35K85: Linear parabolic unilateral problems and linear parabolic variational inequalities [See also 35R35, 49J40]

Keywords
Backward stochastic differential equation probabilistic representation of solution of second order parabolic PDE obstacle problems for second order parabolic PDE

Citation

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. doi:10.1214/aop/1024404416. https://projecteuclid.org/euclid.aop/1024404416


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  • LATP, URA CNRS 225 Centre de Math´ematiques et d'Informatique Universit´e 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