Real time solution of Duncan-Mortensen-Zakai equation without memory

It is well known that the nonlinear filtering problem has important applications in both military and commercial industries. The central problem of nonlinear filtering is to solve the DMZ equation in real time and memoryless manner. The purpose of this paper is to show that, under very mild conditions (which essentially say that the growth of the observation |h| is greater than the growth of the drift |f|), the DMZ equation admits a unique nonnegative weak solution u which can be approximated by a solution uR of the DMZ equation on the ball BR with uR|¿BR = 0. The error of this approximation is bounded by a function of R which tends to zero as R goes to infinity. The solution uR can in turn be approximated efficiently by an algorithm depending only on solving the observation-independent Kolmogorov equation on BR. In theory, our algorithm can solve basically all engineering problems in real time manner. Specifically, we show that the solution obtained from our algorithms converges to the solution of the DMZ equation in L1-sense. Equally important, we have a precise error estimate of this convergence which is important in numerical computation.

[1]  Stephen S.-T. Yau,et al.  Real time solution of nonlinear filtering problem without memory I , 2000 .

[2]  E. Pardouxt,et al.  Stochastic partial differential equations and filtering of diffusion processes , 1980 .

[3]  F. Gland,et al.  Time-discretization of the Zakai equation for diffusion processes observed in correlated noise , 1991 .

[4]  G. Pagès,et al.  Optimal quantization methods for nonlinear filtering with discrete-time observations , 2005 .

[5]  R. Glowinski,et al.  Approximation of the Zakai¨ equation by the splitting up method , 1990 .

[6]  D. Michel,et al.  Régularité des lois conditionnelles en théorie du filtrage non-linéaire et calcul des variations stochastique , 1981 .

[7]  P. Protter,et al.  The Monte-Carlo method for filtering with discrete-time observations , 2001 .

[8]  S. Mitter,et al.  Optimal control and nonlinear filtering for nondegenerate diffusion processes , 1982 .

[9]  B. Rozovskii,et al.  Nonlinear Filtering Revisited: A Spectral Approach , 1997 .

[10]  É. Pardoux,et al.  quations du filtrage non linaire de la prdiction et du lissage , 1982 .

[11]  M. Zakai.  On the optimal filtering of diffusion processes , 1969 .

[12]  Wing Shing Wong,et al.  Lie-Trotter product formulas for nonlinear filtering , 1986 .

[13]  K. Ito.  Approximation of the Zakai Equation for Nonlinear Filtering , 1996 .

[14]  R. E. Kalman,et al.  New Results in Linear Filtering and Prediction Theory , 1961 .

[15]  É. Pardoux,et al.  Un théorème d'unicité pour l'equation de zakaï , 1990 .

[16]  R. Glowinski,et al.  Approximation of some stochastic differential equations by the splitting up method , 1992 .

[17]  Héctor J. Sussmann,et al.  Rigorous Results on the Cubic Sensor Problem , 1981 .

[18]  Kazufumi Ito,et al.  Approximation of the Kushner Equation for Nonlinear Filtering , 2000, SIAM J. Control. Optim..

[19]  Harold J. Kushner,et al.  A Robust Discrete State Approximation to the Optimal Nonlinear Filter for a Diffusion. , 1980 .

[20]  Noriaki Nagase.  Remarks on Nonlinear Stochastic Partial Differential Equations: An Application of the Splitting-Up Method , 1995 .

[21]  I. Gyöngy,et al.  On the splitting-up method and stochastic partial differential equations , 2003 .

[22]  Mark H. A. Davis.  On a multiplicative functional transformation arising in nonlinear filtering theory , 1980 .

[23]  D. Crisan,et al.  A particle approximation of the solution of the Kushner–Stratonovitch equation , 1999 .

[24]  John S. Baras,et al.  Existence, uniqueness, and asymptotic behavior of solutions to a class of Zakai equations with unbounded coefficients , 1983 .

[25]  É. Pardoux,et al.  Filtrage Non Lineaire Et Equations Aux Derivees Partielles Stochastiques Associees , 1991 .