Strong approximation of nonlinear filtering for multiscale McKean-Vlasov stochastic systems

TL;DR

This paper studies the approximation of nonlinear filtering in multiscale McKean-Vlasov stochastic systems, proving convergence of the slow component and its filter to an averaged system with explicit convergence rates.

Contribution

It establishes the strong convergence of the slow component and its nonlinear filter to an averaged system in multiscale McKean-Vlasov stochastic systems, including explicit convergence orders.

Findings

Slow component converges to an averaged system in $L^{2p}$ sense.

Nonlinear filter of the slow component converges to that of the averaged system.

Explicit strong convergence order in the $L^2$ case.

Abstract

This work concerns the nonlinear filtering problem of multiscale McKean-Vlasov stochastic systems where the whole systems depend on distributions of fast components. First of all, we prove that the slow component of the original system converges to an average system in the $L^{2p}$ ($p\geqslant 1$) sense. Moreover, we obtain the strong convergence order for the $L^2$ case. Then, given an observation process which depends on the slow component and its distribution, we show that the nonlinear filtering of the slow component and its distribution also converges to that of the average system in the $L^{q}$ ($p\geq 8, 1\leq q\leq \frac{p}{8}$) sense.