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Register a new userUniqueness and superposition of the distribution-dependent Zakai equations
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The work concerns the Zakai equations from nonlinear filtering problems of McKean-Vlasov stochastic differential equations with correlated noises. First, we establish the Kushner-Stratonovich equations, the Zakai equations and the distribution-dependent Zakai equations. And then, the pathwise uniqueness, uniqueness in joint law and uniqueness in law of weak solutions for the distribution-dependent Zakai equations are shown. Finally, we prove a superposition principle between the distribution-dependent Zakai equations and distribution-dependent Fokker-Planck equations. As a by-product, we give some conditions under which distribution-dependent Fokker-Planck equations have unique weak solutions.
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The work concerns the superposition between the Zakai equations and the Fokker-Planck equations on measure spaces. First, we prove a superposition principle for the Fokker-Planck equations on $mR^mN$ under the integrable condition. And then by means of it, we show two superposition principles for the weak solutions of the Zakai equations from the nonlinear filtering problems and the weak solutions of the Fokker-Planck equations on measure spaces. As a by-product, we give some weak conditions under which the Fokker-Planck equations can be solved in the weak sense.
Due to their intrinsic link with nonlinear Fokker-Planck equations and many other applications, distribution dependent stochastic differential equations (DDSDEs for short) have been intensively investigated. In this paper we summarize some recent progresses in the study of DDSDEs, which include the correspondence of weak solutions and nonlinear Fokker-Planck equations, the well-posedness, regularity estimates, exponential ergodicity, long time large deviations, and comparison theorems.
To characterize the Neumann problem for nonlinear Fokker-Planck equations, we investigate distribution dependent reflecting SDEs (DDRSDEs) in a domain. We first prove the well-posedness and establish functional inequalities for reflecting SDEs with singular drifts, then extend these results to DDRSDEs with singular or monotone coefficients, for which a general criterion deducing the well-posedness of DDRSDEs from that of reflecting SDEs is established. Moreover, three different types of exponential ergodicity are derived for DDRSDEs under dissipative, partially dissipative, and fully non-dissipative conditions respectively.
In this article we introduce a new method for the construction of unique strong solutions of a larger class of stochastic delay equations driven by a discontinuous drift vector field and a Wiener process. The results obtained in this paper can be regarded as an infinite-dimensional generalization of those of A. Y. Veretennikov [42] in the case of certain stochastic delay equations with irregular drift coefficients. The approach proposed in this work rests on Malliavin calculus and arguments of a local time variational calculus, which may also be used to study other types of stochastic equations as e.g. functional It^{o}-stochastic differential equations in connection with path-dependent Kolmogorov equations [15].
We consider conditional McKean-Vlasov stochastic differential equations (SDEs), such as the ones arising in the large-system limit of mean field games and particle systems with mean field interactions when common noise is present. The conditional time-marginals of the solutions to these SDEs satisfy non-linear stochastic partial differential equations (SPDEs) of the second order, whereas the laws of the conditional time-marginals follow Fokker-Planck equations on the space of probability measures. We prove two superposition principles: The first establishes that any solution of the SPDE can be lifted to a solution of the conditional McKean-Vlasov SDE, and the second guarantees that any solution of the Fokker-Planck equation on the space of probability measures can be lifted to a solution of the SPDE. We use these results to obtain a mimicking theorem which shows that the conditional time-marginals of an Ito process can be emulated by those of a solution to a conditional McKean-Vlasov SDE with Markovian coefficients. This yields, in particular, a tool for converting open-loop controls into Markovian ones in the context of controlled McKean-Vlasov dynamics.
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