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Register a new userSemiparametric estimation of McKean-Vlasov SDEs
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Vytaut\\.e Pilipauskait\\.e
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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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.
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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.
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.
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.
We derive independence tests by means of dependence measures thresholding in a semiparametric context. Precisely, estimates of phi-mutual informations, associated to phi-divergences between a joint distribution and the product distribution of its margins, are derived through the dual representation of phi-divergences. The asymptotic properties of the proposed estimates are established, including consistency, asymptotic distributions and large deviations principle. The obtained tests of independence are compared via their relative asymptotic Bahadur efficiency and numerical simulations. It follows that the proposed semiparametric Kullback-Leibler Mutual information test is the optimal one. On the other hand, the proposed approach provides a new method for estimating the Kullback-Leibler mutual information in a semiparametric setting, as well as a model selection procedure in large class of dependency models including semiparametric copulas.
In this paper, utilizing Wangs Harnack inequality with power and the Banach fixed point theorem, the weak well-posedness for distribution dependent SDEs with integrable drift is investigated. In addition, using a trick of decoupled method, some regularity such as relative entropy and Sobolevs estimate of invariant probability measure are proved. Furthermore, by comparing two stationary Fokker-Planck-Kolmogorov equations, the existence and uniqueness of invariant probability measure for McKean-Vlasov SDEs are obtained by log-Sobolevs inequality and Banachs fixed theorem. Finally, some examples are presented.
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