Nonlinear Filtering Theory for McKean--Vlasov Type Stochastic Differential Equations

Abstract

We consider estimation theory for partially observed stochastic dynamical systems whose state equations are given by McKean--Vlasov type stochastic differential equations and hence contain a measure term corresponding to the distribution of the solution of the state process. Nonlinear filtering equations are derived in this framework based upon the classification that the measure term is either stochastic or deterministic and that either the state or the measure term is estimated. When the measure term is deterministic the standard theory holds. Further, when the measure is random, the induced functions in the dynamics of the state become random and a similar recursion for the optimal filter is obtained. The joint estimation of state and the measure term is next considered. The extended state in this setup is shown to be a Banach space valued stochastic process with random functions in its state dynamics and a nonlinear filtering equation for this setup is provided. The first step of such an analysis requires an Itô's lemma for Banach space valued stochastic processes with the given dynamics. This work is motivated by state estimation problems in mean field game theory for systems where both major (asymptotically nonnegligible in population size) agents and minor (asymptotically negligible) agents are present.

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