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Article Dans Une Revue Bernoulli Année : 2020

Penalisation techniques for one-dimensional reflected rough differential equations

Résumé

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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Dates et versions

hal-01982781 , version 1 (15-01-2019)
hal-01982781 , version 2 (26-04-2019)
hal-01982781 , version 3 (15-11-2019)
hal-01982781 , version 4 (12-03-2020)
hal-01982781 , version 5 (01-09-2020)

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Citer

Alexandre Richard, Etienne Tanré, Soledad Torres. Penalisation techniques for one-dimensional reflected rough differential equations. Bernoulli, 2020, 26 (4), pp.2949--2986. ⟨10.3150/20-BEJ1212⟩. ⟨hal-01982781v5⟩
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