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Penalisation techniques for one-dimensional reflected rough differential equations

Abstract : In this paper we solve real-valued rough differential equations (RDEs) reflected on an irregular boundary. The solution $Y$ is constructed as the limit of a sequence $(Y^n)_{n\in\mathbb{N}}$ of solutions to RDEs with unbounded drifts $(\psi_n)_{n\in\mathbb{N}}$. 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 the law at time $t>0$ of some reflected Gaussian RDE is absolutely contiuous with respect to the Lebesgue measure.
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Submitted on : Thursday, March 12, 2020 - 11:52:06 AM
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  • HAL Id : hal-01982781, version 4
  • ARXIV : 1904.11447

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Alexandre Richard, Etienne Tanré, Soledad Torres. Penalisation techniques for one-dimensional reflected rough differential equations. Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, inPress. ⟨hal-01982781v4⟩

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