Abstract
In this paper, we establish concentration inequalities both for functionals of the whole solution on an interval [0, T ] of an additive SDE driven by a fractional Brownian motion with Hurst parameter H ∈ (0, 1) and for functionals of discrete-time observations of this process. Then, we apply this general result to specific functionals related to discrete and continuous-time occupation measures of the process.
Highlights
In this article, we consider the solution (Yt)t≥0 of the following Rd-valued Stochastic Differential Equation (SDE) with additive noise: tYt = x + b(Ys)ds + σBt. (1.1)with B a d-dimensional fractional Brownian motion with Hurst parameter H ∈ (0, 1)
K=1 with X = (Yk∆)1 k n and Y is the solution of (1.1) when B is the classical Brownian motion. This decomposition has inspired the approach described in this paper: instead of proving an L1 transportation inequality (1.2), we prove its equivalent formulation (1.3) by using a similar decomposition and the series expansion of the exponential function
We prove several results under an assumption of contractivity on the drift term b in (1.1)
Summary
We consider the solution (Yt)t≥0 of the following Rd-valued Stochastic Differential Equation (SDE) with additive noise: t. For the discrete-time case, they used a kind of tensorization of the L1 transportation inequality but the Markovian nature of the process was essential They prove T1(C) through its equivalent formulation (1.3) and to this end, they apply a decomposition of the functional in (1.3) into a sum of martingale increments, namely: n. K=1 with X = (Yk∆) k n and Y is the solution of (1.1) when B is the classical Brownian motion This decomposition has inspired the approach described in this paper: instead of proving an L1 transportation inequality (1.2), we prove its equivalent formulation (1.3) by using a similar decomposition and the series expansion of the exponential function.
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