2005/10/31 by Dan Crisan, Jie Xiong, Crisan, Dan +1 · 2 citations
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #35R60 #60H15 #60K35 #93E11 #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.math/0510668
openalex publication_date 2005/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The solution ϑ =(ϑt)t≥ 0 of a class of linear stochastic partial differential equations is approximated using Clark's robust representation approach (\citec, \citecc). The ensuing approximations are shown to coincide with the time marginals of solutions of a certain McKean-Vlasov type equation. We prove existence and uniqueness of the solution of the McKean-Vlasov equation. The result leads to a representation of ϑ as a limit of empirical distributions of systems of equally weighted particles. In particular, the solution of the Zakai equation and that of the Kushner-Stratonovitch equation (the two main equations of nonlinear filtering) are shown to be approximated the empirical distribution of systems of particles that have equal weights (unlike those presented in \citekj1 and \citekj2) and do not require additional correction procedures (such as those introduced in \citedan3, \citedan4, \citedmm, etc).