Malliavin Calculus for rough stochastic differential equations

Fabio Bugini; Michele Coghi; Torstein Nilssen

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arXiv · 2402.12056

Malliavin Calculus for rough stochastic differential equations

Fabio Bugini·Michele Coghi·Torstein Nilssen

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Abstract

In this work we show that rough stochastic differential equations (RSDEs), as introduced by Friz, Hocquet, and L\^e (2021), are Malliavin differentiable. We use this to prove existence of a density when the diffusion coefficients satisfies standard ellipticity assumptions. Moreover, when the coefficients are smooth and the diffusion coefficients satisfies a H\"ormander condition, the density is shown to be smooth. The key ingredient is to develop a comprehensive theory of linear rough stochastic differential equations, which could be of independent interest.

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Fabio Bugini, Michele Coghi, Torstein Nilssen. 2024-02-19. Malliavin Calculus for rough stochastic differential equations. https://arxiv.org/abs/2402.12056

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