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.


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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