Summary
This paper concerns the nonlinear filtering problem of calculating “estimates” E[f(xt)¦y s, s≦t] where {x t} is a Markov process with infinitesimal generator A and {y t} is an observation process given by dy t=h(xt)dt +dwtwhere {w t} is a Brownian motion. If h(xt) is a semimartingale then an unnormalized version of this estimate can be expressed in terms of a semigroup T ys,t obtained by a certain y-dependent multiplicative functional transformation of the signal process {x t}. The objective of this paper is to investigate this transformation and in particular to show that under very general conditions its extended generator is A yt f=ey(t)h(A− 1/2h2)(e−y(t)h f).
Article PDF
Similar content being viewed by others
References
Benveniste, A., Jacod, J.: Systèmes de Lévy des processus de Markov. Invent. Math. 21, 183–198 (1973)
Clark, J.M.C.: The design of robust approximations to the stochastic differential equations of nonlinear filtering, in Communication Systems and Random Process Theory, ed. J.K. Skwirzynski. NATO Advanced Study Institute Series. Alphen aan den Rijn: Sijthoff and Noordhoff 1978
Davis, M.H.A.: A pathwise solution of the equations of nonlinear filtering; to appear in Teor. Veroyatnost. i Primenen.
Davis, M.H.A.: Pathwise solutions and multiplicative functionals in nonlinear filtering. Proc. 18th IEEE Conference on Decision and Control, Ft. Lauderdale, Florida. 1979
Friedman, A.: Partial differential equations of parabolic type. Englewood Cliffs, N.J.: Prentice Hall 1964
Jacod, J.: Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Mathematics 714. Berlin-Heidelberg-New York: Springer 1979
Kallianpur, G., Striebel, C.: Estimation of stochastic processes: arbitrary system process with additive white noise observation errors. Ann. Math. Statist. 39, 785–801 (1968)
Kunita, H., Watanabe, S.: On square integrable martingales. Nagoya Math. J. 30, 209–245 (1967)
Kushner, H.J.: A robust discrete state approximation to the optimal nonlinear filter for a diffusion. Stochastics 3, 75–83 (1979)
Liptser, R.S., Shiryaev, A.N.: Statistics of Random Processes I. Berlin-Heidelberg-New York: Springer 1977
Meyer, P.A.: Integrales stochastiques IV in Séminaire de Probabilités I. Lectures Notes in Mathematics 39. Berlin-Heidelberg-New York: Springer 1967
Meyer, P.A.: Transformation des processus de Markov, in Ecole d'Eté de Probabilités de Saint-Flour III. Lecture Notes in Mathematics 390, Berlin-Heidelberg-New York: Springer 1974
Pardoux, E.: Backward and forward stochastic partial differential equations associated with a nonlinear filtering problem. Proc. 18th IEEE Conference on Decision and Control, Ft. Lauderdale, Florida, 1979
Stroock, D.W.: Diffusion processes associated with Lévy generators. Z. Wahrscheinlichkeitstheorie verw. Gebiete 32, 209–244 (1975)
Stroock, D.W., Varadhan, S.R.S.: Diffusion processes with boundary conditions. Comm. Pure Appl. Math. 24, 147–226 (1971)
Watanabe, S.: On discontinuous additive functionals and Lévy measures of Markov process. Japan J. Math. 34, 53–70 (1964)
Zakai, M.: On the optimal filtering of diffusion processes. Z. Wahrscheinlichkeitstheorie verw. Geb. 11, 230–243 (1969)
Author information
Authors and Affiliations
Additional information
Work partially supported by the U.S. Department of Energy (Contract ET-76-C-01-2295) at the Massachusetts Institute of Technology
Rights and permissions
About this article
Cite this article
Davis, M.H.A. On a multiplicative functional transformation arising in nonlinear filtering theory. Z. Wahrscheinlichkeitstheorie verw Gebiete 54, 125–139 (1980). https://doi.org/10.1007/BF00531444
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00531444