Large deviation principle for reflected diffusion process fractional Brownian motion

Ibrahima Sane

doi:10.31197/atnaa.767867

Abstract

In this paper we establish a large deviation principle for solution of perturbed reflected stochastic differential equations driven by a fractional Brownian motion B^H with Hurst index H ∈ (0;1). The key is to prove a uniform Freidlin-Wentzell estimates of solution on the set of continuous square integrable functions in the dual of Schwartz space . We have built in the whole interval of H ∈ (0;1) a new approch different from that of Y. Inahama [10] for LDP of εBH in [6].Thanks to this we establish the LDP for the process diffusion of reflected stochastic differential equations via the principle of contraction on the set of continuous square integrable functions in the dual of Schwartz space.The existence and uniqueness of the solutions of such equations (1) and (2) are obtained by [7].

Highlights

There are dierent methods to show that the diusion process satises the principle of large deviations (LDP), for which several authors have determined the rate function in dierent spaces
In this paper we establish a large deviation principle for solution of perturbed reected stochastic dierential equations driven by a fractional Brownian motion BtH with Hurst index H ∈ (0; 1)
Zhang ([5]) a new approch via the principle of contraction,in the framework of stochastic dierential equations directed by a fractional Brownian motion of Hurst parameter H ∈ (0; 1),by determining the rate function of the dual of Schwartz space

Summary

Introduction

There are dierent methods to show that the diusion process satises the principle of large deviations (LDP), for which several authors have determined the rate function in dierent spaces. In the case of the large deviation principle for a standart Brownian motion, many autors had established the LDP for perturbed diusion processes see among others L. Inahama ([10]) proved, in the framework of the rough trajectory theory that the process εBtH obeys a large deviation principle for. Zhang ([5]) a new approch via the principle of contraction,in the framework of stochastic dierential equations directed by a fractional Brownian motion (fBm) of Hurst parameter H ∈ (0; 1),by determining the rate function of the dual of Schwartz space.

Preliminaries

Main Results

Large deviation principle for reected fractional diusion process

Large deviation principle for perturbed fractional diusion process

Conclusion