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April 1997 Reflected solutions of backward SDE's, and related obstacle problems for PDE's
N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng, M. C. Quenez
Ann. Probab. 25(2): 702-737 (April 1997). DOI: 10.1214/aop/1024404416


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.


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N. El Karoui. C. Kapoudjian. E. Pardoux. S. Peng. M. C. Quenez. "Reflected solutions of backward SDE's, and related obstacle problems for PDE's." Ann. Probab. 25 (2) 702 - 737, April 1997.


Published: April 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0899.60047
MathSciNet: MR1434123
Digital Object Identifier: 10.1214/aop/1024404416

Primary: 35K85 , 60H10 , 60H30

Keywords: backward stochastic differential equation , obstacle problems for second order parabolic PDE , probabilistic representation of solution of second order parabolic PDE

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 2 • April 1997
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