No Arabic abstract
Various particle filters have been proposed over the last couple of decades with the common feature that the update step is governed by a type of control law. This feature makes them an attractive alternative to traditional sequential Monte Carlo which scales poorly with the state dimension due to weight degeneracy. This article proposes a unifying framework that allows to systematically derive the McKean-Vlasov representations of these filters for the discrete time and continuous time observation case, taking inspiration from the smooth approximation of the data considered in Crisan & Xiong (2010) and Clark & Crisan (2005). We consider three filters that have been proposed in the literature and use this framework to derive It^{o} representations of their limiting forms as the approximation parameter $delta rightarrow 0$. All filters require the solution of a Poisson equation defined on $mathbb{R}^{d}$, for which existence and uniqueness of solutions can be a non-trivial issue. We additionally establish conditions on the signal-observation system that ensures well-posedness of the weighted Poisson equation arising in one of the filters.
We study the optimal control of path-dependent McKean-Vlasov equations valued in Hilbert spaces motivated by non Markovian mean-field models driven by stochastic PDEs. We first establish the well-posedness of the state equation, and then we prove the dynamic programming principle (DPP) in such a general framework. The crucial law invariance property of the value function V is rigorously obtained, which means that V can be viewed as a function on the Wasserstein space of probability measures on the set of continuous functions valued in Hilbert space. We then define a notion of pathwise measure derivative, which extends the Wasserstein derivative due to Lions [41], and prove a related functional It{^o} formula in the spirit of Dupire [24] and Wu and Zhang [51]. The Master Bellman equation is derived from the DPP by means of a suitable notion of viscosity solution. We provide different formulations and simplifications of such a Bellman equation notably in the special case when there is no dependence on the law of the control.
In this paper, we present a generic methodology for the efficient numerical approximation of the density function of the McKean-Vlasov SDEs. The weak error analysis for the projected process motivates us to combine the iterative Multilevel Monte Carlo method for McKean-Vlasov SDEs cite{szpruch2019} with non-interacting kernels and projection estimation of particle densities cite{belomestny2018projected}. By exploiting smoothness of the coefficients for McKean-Vlasov SDEs, in the best case scenario (i.e $C^{infty}$ for the coefficients), we obtain the complexity of order $O(epsilon^{-2}|logepsilon|^4)$ for the approximation of expectations and $O(epsilon^{-2}|logepsilon|^5)$ for density estimation.
In this paper we study the problem of semiparametric estimation for a class of McKean-Vlasov stochastic differential equations. Our aim is to estimate the drift coefficient of a MV-SDE based on observations of the corresponding particle system. We propose a semiparametric estimation procedure and derive the rates of convergence for the resulting estimator. We further prove that the obtained rates are essentially optimal in the minimax sense.
By refining a recent result of Xie and Zhang, we prove the exponential ergodicity under a weighted variation norm for singular SDEs with drift containing a local integrable term and a coercive term. This result is then extended to singular reflecting SDEs as well as singular McKean-Vlasov SDEs with or without reflection. We also present a general result deducing the uniform ergodicity of McKean-Vlasov SDEs from that of classical SDEs. As an application, the $L^1$-exponential convergence is derived for a class of non-symmetric singular granular media equations.
The following type exponential convergence is proved for (non-degenerate or degenerate) McKean-Vlasov SDEs: $$W_2(mu_t,mu_infty)^2 +{rm Ent}(mu_t|mu_infty)le c {rm e}^{-lambda t} minbig{W_2(mu_0, mu_infty)^2,{rm Ent}(mu_0|mu_infty)big}, tge 1,$$ where $c,lambda>0$ are constants, $mu_t$ is the distribution of the solution at time $t$, $mu_infty$ is the unique invariant probability measure, ${rm Ent}$ is the relative entropy and $W_2$ is the $L^2$-Wasserstein distance. In particular, this type exponential convergence holds for some (non-degenerate or degenerate) granular media type equations generalizing those studied in [CMV, GLW] on the exponential convergence in a mean field entropy.