Skip to main content
Log in

An inverse problem for stochastic differential equations

  • Articles
  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We discuss the problem of reconstructing the drift coefficient of a diffusion from the knowledge of the transition probabilities outside a given bounded region in ℝd,d>1. We also give an interpretation of the solution of this inverse problem in the framework of stochastic mechanics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. S. Albeverio, Some points of interaction between stochastic analysis and quantum theory, inStochastic Differential Systems, N. Christopeit, K. Helmes, and M. Kohlmann, eds. (Springer, Berlin, 1986), pp. 1–26.

    Google Scholar 

  2. S. Albeverio, Ph. Blanchard, F. Gesztesy, and L. Streit, Quantum mechanical low energy scattering in terms of diffusion processes, inStochastic Aspects of Classical and Quantum Systems, S. Albeverio, Ph. Combe, and M. Sirugue-Collin (Springer, Berlin, 1984), pp. 207–227.

    Google Scholar 

  3. S. Albeverio, Ph. Blanchard, M. Hazewinkel, and L. Streit, eds.,Stochastic Processes in Physics and Engineering (D. Reidel, 1988).

  4. S. Albeverio, R. Høegh-Krohn, and L. Streit, Energy forms, Hamiltonian and distorted Brownian Paths,J. Math. Phys. 18:907 (1977).

    Google Scholar 

  5. G. Alessandrini, On the identification of the leading coefficient of an elliptic equation,Boll. Un. Mat. Ital. C (6)1985:1–25.

  6. L. Arnold,Stochastic Differential Equations: Theory and Applications (Wiley, New York, 1974).

    Google Scholar 

  7. R. Azencottet al., Géodesiques et diffusions en temps petit,Astérisque 1981:84–85.

  8. Ph. Blanchard, Ph. Combe, and W. Zheng, Physical and mathematical aspects of stochastic mechanics,Lecture Notes in Physics, Vol. 281 (Springer, Berlin, 1987).

    Google Scholar 

  9. E. Carlen, Existence and sample path properties of the diffusion process in Nelson's stochastic mechanics, inStochastic Processes in Mathematics and Physics I, S. Albeverio, Ph. Blanchard, and L. Streit, eds. (Springer, Berlin, 1986), pp. 25–51.

    Google Scholar 

  10. G. Dohnal, On estimating the diffusion coefficient,J. Appl. Prob. 24:105–114 (1987).

    Google Scholar 

  11. J. G. B. Beumee, H. Rabitz, An application of filtering theory to parameter identification using stochastic mechanics,J. Math. Phys. 28:1787–1794 (1987).

    Google Scholar 

  12. M. H. A. Davis, Stochastic control and nonlinear filtering, Tata Institute, Bombay (1984).

    Google Scholar 

  13. A. Friedman and B. Gustaffson, Identification of the conductivity coefficient in an elliptic equation,Siam J. Math. Anal. 18:777–787 (1987).

    Google Scholar 

  14. J. Glimm and A. Jaffe,Quantum Physics, A Functional Integral Point of View (Springer, New York, 1981).

    Google Scholar 

  15. N. Ikeda and S. Watanabe,Stochastic Differential Equation and Diffusion Processes (North-Holland, 1981).

  16. S. Kotani, One dimensional random Schrödinger operators and Herglotz function, inProceedings Taniguchi Symposium 1985, Probabilistic methods in mathematical physics (Academic Press, Boston, 1987), pp. 219–250.

    Google Scholar 

  17. W. Loges, Estimation of parameters for Hilbert space-valued partially observable stochastic processes,J. Multivariate Anal. 20:161–174 (1986).

    Google Scholar 

  18. E. Nelson,Quantum Fluctuations (Princeton University Press, 1985).

  19. B. Øksendahl,Stochastic Differential Equations (Springer, Berlin, 1985).

    Google Scholar 

  20. J. Pöschel and E. Trubowitz,Inverse Spectral Theory (Academic Press, Boston, 1987).

    Google Scholar 

  21. L. G. Rogers and D. Williams,Diffusion, Markov Processes and Martingales, Vol. 2,Ito Calculus (Wiley, Chichester, 1987).

    Google Scholar 

  22. L. Streit, Quantum theory and stochastic processes-Some contact points, inStochastic Processes and Their Applications, K. Ito and T. Hida, eds. (Springer, Berlin, 1986), pp. 197–213.

    Google Scholar 

  23. P. C. Sabatier, ed.,Inverse Problems (Academic Press, 1987).

  24. D. S. Schucker, Stochastic mechanics of systems with zero potential,J. Funct. Anal. 38:146 (1980).

    Google Scholar 

  25. B. Simon,Functional Integration and Quantum Physics (Academic Press, New York, 1979).

    Google Scholar 

  26. S. R. S. Varadhan,Lectures on Diffusion Problems and Partial Differential Equations (Springer-Verlag, 1980).

  27. E. Wong and B. Hajek,Stochastic Processes in Engineering Systems (Springer, New York, 1985).

    Google Scholar 

  28. S. Albeverio, K. Yasue, J. C. Zambrini, Euclidean quantum mechanics: analytic approach, Bochum preprint, to appear inAnn. Inst. H. Poincaré (Phys. Th.) (1989).

Download references

Author information

Authors and Affiliations

Authors

Additional information

This paper is dedicated to the dear memory of Paola Calderoni.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Albeverio, S., Blanchard, P., Kusuoka, S. et al. An inverse problem for stochastic differential equations. J Stat Phys 57, 347–356 (1989). https://doi.org/10.1007/BF01023648

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01023648

Key words

Navigation