Analytical approximations of non-linear SDEs of McKean–Vlasov type

Abstract

We provide analytical approximations for the law of the solutions to a certain class of scalar McKean-Vlasov stochastic differential equations (MKV-SDEs) with random initial datum. "Propagation of chaos" results ([Szn91]) connect this class of SDEs with the macroscopic limiting behavior of a particle, evolving within a mean-field interaction particle system, as the total number of particles tends to infinity. Here we assume the mean-field interaction only acting on the drift of each particle, this giving rise to a MKV-SDE where the drift coefficient depends on the law of the unknown solution. By perturbing the non-linear forward Kolmogorov equation associated to the MKV-SDE, we perform a two-steps approximating procedure that decouples the McKean-Vlasov interaction from the standard dependence on the state-variables. The first step yields an expansion for the marginal distribution at a given time, whereas the second yields an expansion for the transition density. Both the approximating series turn out to be asymptotically convergent in the limit of short times and small noise, the convergence order for the latter expansion being higher than for the former. Concise numerical tests are presented to illustrate the accuracy of the resulting approximation formulas. The latter are expressed in semi-closed form and can be then regarded as a viable alternative to the numerical simulation of the large-particle system, which can be computationally very expensive. Moreover, these results pave the way for further extensions of this approach to more general dynamics and to high-dimensional settings.

Mixing time guarantees for unadjusted Hamiltonian Monte Carlo

“…Since we avoid Metropolis-adjustment, time step or duration adaptivity can be easily included in (2.13) -as in [41,8,48,42,78]. For promising alternative strategies to the particle approximation considered here, see [29,31].…”

A Fourier-based Picard-iteration approach for a class of McKean–Vlasov SDEs with Lévy jumps

“…Now set φ t (ξ) := E[Z t ]. By the martingale property of the Itô and the jump integrals, combined with Fubini's theorem, we obtain that φ t (ξ) = 1 + t 0 φ s (ξ)ψ s (ξ)ds, whereψ s (ξ) = R d e i ξ,Φ −1 0,s y − 1 − 1 {|y|<1} i ξ, Φ −1 0,s y ν(dy) + i ξ, Φ −1 0,s β s − 1 2 ξ, Φ −1 0,s σ Φ −1 0,s σ ξ .By differentiating both terms, we have thatd dt φ t (ξ) = φ t (ξ)ψ t (ξ), t > 0, φ 0 (ξ) = 1, which yields that E exp i ξ,X t − Y = φ t (ξ) = e t 0 ψs(ξ)ds ,which in turn, combined with (63), yields(8) and concludes the proof.…”

“…Sun et al [20] developed Itô-Taylor schemes of Euler-and Milstein-type for numerically estimating the solution of MK-V SDEs with Lipschitz regular coefficients and square-integrable initial law. Gobet and Pagliarani [8] recently developed analytical approximations of the transition density of the solutions by extending a perturbation technique that was previously developed for standard SDEs. In [4], Chaudru De Raynal and Garcia Trillos developed a cubature method to obtain estimates for the solution of forward-backward SDEs of MK-V type.…”

Fast and precise inference on diffusivity in interacting particle systems

“…The work in this paper pertaining to analytical expressions of approximate transition densities is in itself not new; there exists a wide literature on the subject that cover many different levels of approximation. See for example (Gobet and Pagliarani 2018 ) for a comprehensive treatise. What is lacking in the literature, however, is simple methods for deriving and using such transitions when facing realistic scientific problems, and thus our contribution is to provide a bridge between the field of stochastic calculus and mathematical biology.…”