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Unformatted text preview: Nonlinear Filtering: Separation of Parameters and Observations Using Galerkin Approximation and Wiener Chaos Decomposition C. P. Fung * S. Lototsky † Published as IMA Preprint Series # 1458, February 1997 Abstract The nonlinear filtering problem is considered for the time homogeneous dif- fusion model. An algorithm is proposed for computing a recursive approxima- tion of the unnormalized filtering density, and the error of the approximation is estimated. The algorithm can have high resolution in time and, unlike most existing algorithms for nonlinear filtering, can be implemented in real time for large dimensions of the sate process. The on-line part of the algorithm is simplified by performing the most time consuming operations off line. 1 Introduction In many problems of stochastic analysis it is necessary to find the best mean square estimate of moments or other similar functionals of a partially observed diffusion process. Assume that X = X ( t ) , t ≥ , is the unobservable component and the observable component Y = Y ( t ) , t ≥ , is given by Y ( t ) = Z t h ( X ( s )) ds + W ( t ) , where W = W ( t ) ,t ≥ , is a Wiener process independent of the process X . If f = f ( x ) is a measurable function satisfying E | f ( X ( t )) | 2 < ∞ , t ≥ , then it is known [11, 13, 20] that under certain regularity assumptions the best mean square estimate ˆ f t of f ( X ( t )) given the trajectory Y ( s ) , s ≤ t, can be written as ˆ f t = R f ( x ) u ( t,x ) dx R u ( t,x ) dx , * Center for Applied Mathematical Sciences, University of Southern California, Los Angeles, CA 90089-1113 ( [email protected] ). † Institute for Mathematics and its Applications, University of Minnesota, 514 Vincent Hall, 206 Church Street S.E., Minneapolis, MN 55455 ( [email protected] ). This work was partially supported by the ONR Grant #N00014-95-1-0229. 1 where u = u ( t,x ) is a random field called the unnormalized filtering density (UFD). The problem of estimating f ( X ( t )) is thus reduced to the problem of computing the UFD u . It is also known that u = u ( t,x ) is the solution of a certain stochastic partial differential equation, called the Zakai equation, driven by the observation process. The exact solution of the Zakai equation can be found only in a few special cases, and as a result the central part of the general nonlinear filtering problem is the numerical solution of the equation. In some applications, e.g. target tracking, the solution must be computed in real time, which puts additional restriction on the corresponding numerical scheme. Most of the existing numerical schemes for the Zakai equation use various gen- eralizations of the corresponding algorithms for the deterministic partial differential equations and therefore cannot be implemented in real time when the dimension of the state process is more than three because of the large amount of computations....
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This note was uploaded on 06/10/2008 for the course MATH 494 taught by Professor Senkevitch during the Fall '07 term at USC.
- Fall '07