Penalisation techniques for one-dimensional reflected rough differential equations

Abstract : In this paper we solve real-valued rough differential equations (RDEs) reflected on a rough boundary. The solution $Y$ is constructed as the limit of a sequence $(Y^n)_{n\in\N}$ of solutions to RDEs with unbounded drifts $(\psi_n)_{n\in\N}$. The penalisation $\psi_n$ increases with $n$. Along the way, we thus also provide an existence theorem and a Doss-Sussmann representation for RDEs with a drift growing at most linearly. In addition, a speed of convergence of the sequence of penalised paths to the reflected solution is obtained. \\ We finally use the penalisation method to prove that under some conditions, the law of a reflected Gaussian RDE at time $t>0$ is absolutely contiuous with respect to the Lebesgue measure.
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https://hal.archives-ouvertes.fr/hal-01982781
Contributor : Alexandre Richard <>
Submitted on : Friday, April 26, 2019 - 8:18:01 AM
Last modification on : Sunday, April 28, 2019 - 1:16:08 AM

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  • HAL Id : hal-01982781, version 2
  • ARXIV : 1904.11447

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Alexandre Richard, Etienne Tanré, Soledad Torres. Penalisation techniques for one-dimensional reflected rough differential equations. 2019. ⟨hal-01982781v2⟩

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