The Annals of Probability
- Ann. Probab.
- Volume 25, Number 2 (1997), 702-737.
Reflected solutions of backward SDE's, and related obstacle problems for PDE's
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.
Ann. Probab., Volume 25, Number 2 (1997), 702-737.
First available in Project Euclid: 18 June 2002
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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