Applied Mathematics and Optimization

, Volume 4, Issue 1, pp 329–346

Exit probabilities and optimal stochastic control


  • Wendell H. Fleming
    • Department of MathematicsBrown University
    • Lefschetz Center for Dynamical Systems, Division of Applied MathematicsBrown University

DOI: 10.1007/BF01442148

Cite this article as:
Fleming, W.H. Appl Math Optim (1977) 4: 329. doi:10.1007/BF01442148


This paper is concerned with Markov diffusion processes which obey stochastic differential equations depending on a small parameterε. The parameter enters as a coefficient in the noise term of the stochastic differential equation. The Ventcel-Freidlin estimates give asymptotic formulas (asε→0) for such quantities as the probability of exit from a regionD through a given portionN of the boundary ∂D, the mean exit time, and the probability of exit by a given timeT. A new method to obtain such estimates is given, using ideas from stochastic control theory.

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© Springer-Verlag New York, Inc 1978