Abstract
The problem of forcing a nondegenerate diffusion process to a given final configuration is considered. Using the logarithmic transformation approach developed by Fleming, it is shown that the perturbation of the drift suggested by Jamison solves an optimal stochastic control problem. Such perturbation happens to have minimum energy between all controls that “bring” the diffusion to the desired final distribution. A special property of the change of measure on the path-space that corresponds to the aforesaid perturbation of the drift is also shown.
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References
H. Akaike, Canonical correlation analysis of time series and the use of an information criterion, in System Identification and Case Studies, R. K. Mehra and D. G. Lainiotis, Eds., Mathematics in Science and Engineering, Vol. 126, Academic Press, New York, 1976, pp. 27–96.
D. G. Arnoson and J. Serrin, Local behaviour of quasilinear parabolic equations, Arch. Rational Mech. Anal. 25, 1967, 81–122.
S. Bernstein, Sur les liasons entre les grandeurs aleatories, Verh. des intern. Mathematikerkongr., Vol. 1, Zurich, 1932, pp. 288–309.
A. Beurling, An automorphism of product measures, Ann. of Math. 72, 1960, 189–200.
R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory, Academic Press, New York, 1968.
P. Dai Pra and M. Pavon, On the Markov process of Schrodinger, the Feynmann—Kac formula and stochastic control, Proceedings of the M.T.N.S. Conference, Amsterdam, 1989 (to appear).
M. H. A. Davis, Detection, mutual information and feedback encoding: applications of stochastic calculus, in Communication System and Random Process Theory, NATO Advanced Study Institutes Series, Vol. 25, 1978, pp. 705–720.
D. Durr and A. Bach, The Onsager—Machlup functional as Lagrangian for the most probable path of a diffusion process, Comm. Math. Phys. 60, 1978, 153–170.
E. B. Dynkin, Markov Processes, Vol. II, Academic Press, New York, 1965.
W. H. Fleming, Exit probabilities and optimal stochastic control, Appl. Math. Optim. 4, 1978, 329–346.
W. H. Fleming, Logarithmic transformations and stochastic control, in Advances in Filtering and Optimal Stochastic Control, W. H. Fleming and L. G. Gorostiza Eds., Lecture Notes in Control and Information Sciences, Vol. 42, Springer-Verlag, Berlin, 1982, pp. 131–141.
W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, 1975.
W. H. Fleming and S. J. Sheu, Stochastic variational formula for fundamental solutions of parabolic PDE, Appl. Math. Optim. 13, 1985, 193–204.
A. Friedmann, Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Cliffs, NJ, 1964.
A. Friedmann, Stochastic Differential Equations and Applications, Academic Press, New York, 1975.
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Kodanska-Wiley, New York, 1980.
B. Jamison, Reciprocal processes, Z. Wahrsch. Vern. Gebiete 30, 1974, 65–86.
B. Jamison, The Markov processes of Schrödinger, Z. Wahrsch. Vern. Gebiete 32, 1975, 323–331.
D. Jamison, Reciprocal processes: the stationary Gaussian case, Ann. Math. Statist. 41, 1970, 1624–1630.
S. Karlin and H. Taylor, A Second Course in Stochastic Processes, Academic Press, New York, 1981.
A. J. Krener, Reciprocal Processes and the Stochastic Realization Problem for Acausal Systems in Modeling, Identification and Robust Control, C. I. Byrnes and A. Lindquist, Eds., North-Holland, Amsterdam, 1986, pp. 197–211.
A. J. Krener, Realizations of reciprocal processes, Modeling and Adaptive Control (Sopron, 1986), Lecture Notes in Control and Information Sciences, Vol. 105, Springer-Verlag, Berlin, 1988, pp. 159–174.
A. J. Krener, Reciprocal Diffusion and Stochastic Differential Equations of Second Order, Stochastics 24, 393–422.
S. Kullback, Information Theory and Statistics, Dover, New York, 1968.
H. H. Kuo, Gaussian Measures in Banach Spaces, Lectures Notes in Mathematics, Vol. 463, Springer-Verlag, Berlin, 1975.
M. Loeve, Probability Theory, Van Nostrand, New York, 1963.
E. Schrödinger, Uber die Umkehrung der Naturgesetze, Sitzung ber Preuss. Akad. Wissen., Berlin Phys. Math., Vol. 144, 1931.
D. W. Strook, Topics in Stochastic Differential Equations, Tata Institute of Fundamental Research, Bombay, Springer-Verlag, New York, 1982.
D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, New York, 1979.
Y. Takahashi and S. Watanabe, The Probability Functionals (Onsager—Machlup Functions) of Diffusion Processes, Lecture Notes in Mathematics, Vol. 851, Springer-Verlag, Berlin, 1980, pp. 433–463.
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Communicated by W. H. Fleming
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Dai Pra, P. A stochastic control approach to reciprocal diffusion processes. Appl Math Optim 23, 313–329 (1991). https://doi.org/10.1007/BF01442404
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DOI: https://doi.org/10.1007/BF01442404