Exit probabilities and optimal stochastic control

Abstract

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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References

  1. 1.

    J. M. Bismut, Théorie probabiliste du contrôle des diffusions,Memoirs Amer. Math. Soc., 167, 1976.

  2. 2.

    M. H. A. Davis and P. Varaiya, Dynamic programming conditions for partially observable stochastic systems,SIAM J. Control, 11, 226–261 (1973).

    Google Scholar 

  3. 3.

    W. H. Fleming, The Cauchy problem for a nonlinear first-order partial differential equation,J. Differential Eqns. 5, 515–530 (1969).

    Google Scholar 

  4. 4.

    W. H. Fleming, Stochastic control for small noise intensities,SIAM J. Control, 9, 473–517 (1971).

    Google Scholar 

  5. 5.

    W. H. Fleming, Inclusion probability and optimal stochastic control,IRIA Seminars Review, 1977.

  6. 6.

    W. H. Fleming and R. Rishel,Deterministic and Stochastic Optimal Control, Springer-Verlag, 1975.

  7. 7.

    A. Friedman,Stochastic Differential Equations and Applications, Academic Press, vol. I, 1975; vol. II, 1976.

  8. 8.

    C. J. Holland, A new energy characterization of the smallest eigenvalue of the Schrödinger equation,Communications Pure Appl. Math., 30, 755–765 (1977).

    Google Scholar 

  9. 9.

    C. J. Holland, A minimum principle for the principal eigenvalue for second order linear elliptic equations with natural boundary conditions,Communications Pure Appl. Math., (in press).

  10. 10.

    E. Hopf, The partial differential equationu t + uux = µuxx,Communications Pure Appl. Math., 3, 201–230 (1950).

    Google Scholar 

  11. 11.

    O. A. Ladyzhenskaya and N. N. Uralseva,Linear and Quasilinear Elliptic Equations, Academic Press, 1968.

  12. 12.

    O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uralseva,Linear and Quasilinear Equations of Parabolic Type, American Math. Soc., 1968.

  13. 13.

    P. Priouret, Ecole d'Eté de Probabilités de Saint-Flour III,Springer-Lecture Notes in Math., 390, Springer-Verlag, 1974.

  14. 14.

    A. V. Skorokhod, Limit theorems for stochastic processes,Theory Prob. Appl. 1, 261–290 (1956).

    Google Scholar 

  15. 15.

    A. D. Ventcel, Rough limit theorems on large deviations for Markov stochastic processes,Theory Prob. Appl., 21, 227–242, 499–512 (1976).

    Google Scholar 

  16. 16.

    A. D. Ventcel and M. I. Freidlin, On small random perturbations of dynamical systems,Russian Math. Surveys 25, 1–56 (1970).

    Google Scholar 

  17. 17.

    A. D. Ventcel and M. I. Freidlin, Some problems concerning stability under small random perturbations,Theory Prob. Appl., 17, 269–283 (1972).

    Google Scholar 

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This research was supported by the Air Force Office of Scientific Research under AF-AFOSR 76-3063, and in part by the National Science Foundation under NSF-MCS 76-37247.

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Fleming, W.H. Exit probabilities and optimal stochastic control. Appl Math Optim 4, 329–346 (1977). https://doi.org/10.1007/BF01442148

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Keywords

  • Differential Equation
  • System Theory
  • Diffusion Process
  • Mathematical Method
  • Control Theory