182 research outputs found
On small time asymptotics for rough differential equations driven by fractional Brownian motions
We survey existing results concerning the study in small times of the density
of the solution of a rough differential equation driven by fractional Brownian
motions. We also slightly improve existing results and discuss some possible
applications to mathematical finance.Comment: This is a survey paper, submitted to proceedings in the memory of
Peter Laurenc
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, c\ue0dl\ue0g noise,
and reflecting boundaries. Last not least, we generalize Dawson-G\ue4rtner large
deviations and the central limit theorem to a non-Brownian noise setting
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