Pathwise McKean–Vlasov theory with additive noise

Abstract

We take a pathwise approach to classical McKean-Vlasov stochastic differential equations with additive noise, as e.g. exposed in Sznitmann [38]. Our study was prompted by some concrete problems in battery modelling [23], and also by recent progrss on rough-pathwise McKean-Vlasov theory, notably Cass-Lyons [10], and then Bailleul, Catellier and Delarue [4]. Such a "pathwise McKean-Vlasov theory" can be traced back to Tanaka [40]. This paper can be seen as an attempt to advertize the ideas, power and simplicity of the pathwise appproach, not so easily extracted from [4,10,40], together with a number of novel applications. These include mean field convergence without a priori independence and exchangeability assumption; common noise, càdlàg noise, and reflecting boundaries. Last not least, we generalize Dawson-Gärtner large deviations and the central limit theorem to a non-Brownian noise setting.

Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering

“…Reiterating the discussion so far, solutions to (25)- (26) provide exact solutions to the filtering problem (4), while solutions to (23)-( 24) lead to approximate ones (except in the linear Gaussian case). However, as P is given explicitly in (24), the system (23)- (24) lends itself straightforwardly to efficient numerical integration, while the system (25)-(26) poses a formidable numerical challenge in the form of the high-dimensional PDE (26). What is more, well-posedness of systems of the type ( 25)- (26), with coefficients that depend on the law through the solution of a PDE is currently not well understood.…”

“…What is more, well-posedness of systems of the type ( 25)- (26), with coefficients that depend on the law through the solution of a PDE is currently not well understood. Nevertheless, McKean-Vlasov formulations of the type ( 25)- (26) conveniently link between the theoretically optimal Kushner-Stratonovich SPDE (19) and the numerically tractable and practically relevant ensemble Kalman dynamics ( 23)- (24). In this paper, we leverage this viewpoint in order to construct a robust version of (17).…”

“…The equation is then solved as a fixed-point in the mixed R d and L p -space. In [24] the case of pathwise McKean-Vlasov equation with additive noise is considered. The basic techniques used are similar to the ones used in the rough-path case, but the need for rough paths is removed thanks to the additive noise.…”

Propagation of chaos for mean field rough differential equations

“…Remark 4.4. Recently, the authors in [13] obtained a quantified propagation of chaos result for mean field stochastic equations with additive noise…”

Distribution dependent SDEs driven by additive continuous noise

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We study distribution dependent stochastic differential equation driven by a continuous process, without any specification on its law, following the approach initiated in [16]. We provide several criteria for existence and uniqueness of solutions which go beyond the classical globally Lipschitz setting.

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