TY - JOUR

T1 - Fixed-point smoothing of scalar diffusions II

T2 - The error of the optimal smoother

AU - Steinberg, Y.

AU - Bobrovsky, B. Z.

AU - Schuss, Z.

PY - 2000

Y1 - 2000

N2 - The problem of fixed-point smoothing of a scalar diffusion process consists of estimating the initial value of the process, given its noisy measurements as a function of time. An asymptotic expansion of the joint filtering-smoothing conditional density function is constructed in the limit of small measurements noise. The approximate optimal nonlinear fixed-point smoother of Part I [SIAM J. Appl. Math., 54 (1994), pp. 833-853] is rederived from the expansion. A detailed analysis of the conditional mean square estimation error (CMSEE) of the optimal fixed-point smoother and of its leading-order approximation is presented. It is shown that if the initial error is small, e.g., if asymptotically optimal filtering is used first, the leading-order approximation to the optimal smoother is three dimensional and thus simpler than the four-dimensional extended Kalman smoother. Furthermore, nonlinear fixed-point smoothing can reduce the CMSEE relative to that of filtering by a factor of 1/2 within smoothing time proportional to the noise-intensity parameter. If the initial error is not small, it is shown that even in the linear case the CMSEE of the optimal fixed-point smoother is asymptotically the same as that of the optimal filter, in this limit.

AB - The problem of fixed-point smoothing of a scalar diffusion process consists of estimating the initial value of the process, given its noisy measurements as a function of time. An asymptotic expansion of the joint filtering-smoothing conditional density function is constructed in the limit of small measurements noise. The approximate optimal nonlinear fixed-point smoother of Part I [SIAM J. Appl. Math., 54 (1994), pp. 833-853] is rederived from the expansion. A detailed analysis of the conditional mean square estimation error (CMSEE) of the optimal fixed-point smoother and of its leading-order approximation is presented. It is shown that if the initial error is small, e.g., if asymptotically optimal filtering is used first, the leading-order approximation to the optimal smoother is three dimensional and thus simpler than the four-dimensional extended Kalman smoother. Furthermore, nonlinear fixed-point smoothing can reduce the CMSEE relative to that of filtering by a factor of 1/2 within smoothing time proportional to the noise-intensity parameter. If the initial error is not small, it is shown that even in the linear case the CMSEE of the optimal fixed-point smoother is asymptotically the same as that of the optimal filter, in this limit.

KW - Filtering

KW - Singular perturbations

KW - Stochastic differential equations

UR - http://www.scopus.com/inward/record.url?scp=0034845440&partnerID=8YFLogxK

U2 - 10.1137/s0036139997321220

DO - 10.1137/s0036139997321220

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AN - SCOPUS:0034845440

SN - 0036-1399

VL - 61

SP - 1431

EP - 1444

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

IS - 4

ER -