New explicit filters and smoothers for diffusions with nonlinear drift and measurements

✍ Scribed by Charalambos D. Charalambous; Robert J. Elliott

• Book ID: 104301094
• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 748 KB
• Volume: 33
• Category: Article
• ISSN: 0167-6911
• DOI: 10.1016/s0167-6911(97)00095-9

✦ Synopsis

The optimal least-sqaares filtering of a diffusion x(t) from its noisy measurements {y(z); 0 ~< r ~< t} is given by the conditional mean E[x(t)ly(z); 0 <~ z <~ t]. When x(t) satisfies the stochastic diffusion equation dx(t)= f(x(t))dt + dw(t) and y(t)= foX(S)ds + b(t), where f(.) is a global solution of the Riccati equation 6~/Oxf(x)+ f(x)2 =~x2 + fix + 7, for some (~,fl, 7)E~R 3, and w(-), b(.) are independent Brownian motions, Benes gave an explicit formula for computing the conditional mean. This paper extends Benes results to measurements y(t): foX(S)ds ÷ fo dx(s)÷ b(t) (and its multidimensiorLal version) without imposing additional conditions on f(.). Analogous results are also derived for the optimal least-sc~uares smoothed estimate E[x(s) ly(z); 0 <~ z <~ t], s < t. The methodology relies on Girsanov's measure transformation:~, gauge transformations, function space integrations, Lie algebras, and the Duncan-Mortensen-Zakai equation. (~) 1993 Elsevier Science B.V. All rights reserved.