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November 2020 Penalisation techniques for one-dimensional reflected rough differential equations
Alexandre Richard, Etienne Tanré, Soledad Torres
Bernoulli 26(4): 2949-2986 (November 2020). DOI: 10.3150/20-BEJ1212

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 continuous with respect to the Lebesgue measure.

Citation

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Alexandre Richard. Etienne Tanré. Soledad Torres. "Penalisation techniques for one-dimensional reflected rough differential equations." Bernoulli 26 (4) 2949 - 2986, November 2020. https://doi.org/10.3150/20-BEJ1212

Information

Received: 1 April 2019; Revised: 1 November 2019; Published: November 2020
First available in Project Euclid: 27 August 2020

zbMATH: 07256165
MathSciNet: MR4140534
Digital Object Identifier: 10.3150/20-BEJ1212

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

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Vol.26 • No. 4 • November 2020
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