Penalisation techniques for one-dimensional reflected rough differential equations

Alexandre Richard (1, 2) , Etienne Tanré (3) , Soledad Torres (4)

1 MICS - Mathématiques et Informatique pour la Complexité et les Systèmes
2 FR3487 - Fédération de Mathématiques de l'Ecole Centrale Paris
3 TOSCA - TO Simulate and CAlibrate stochastic models
4 CIMFAV - Centro de Investigación y Modelamiento de Fenómenos Aleatorios – Valparaíso

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.

Mots clés: Reected rough dierential equation; Penalisation; Gaussian noise; Skorokhod problem MSC2010 Subject Classication: 34F05; 60G15; 60H10

Domaines: Probabilités [math.PR]

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