Recursive computation of the invariant measure of a stochastic differential equation driven by a Lévy process

Abstract

We study some recursive procedures based on exact or approximate Euler schemes with decreasing step to compute the invariant measure of Lévy driven SDEs. We prove the convergence of these procedures toward the invariant measure under weak conditions on the moment of the Lévy process and on the mean-reverting of the dynamical system. We also show that an a.s. CLT for stable processes can be derived from our main results. Finally, we illustrate our results by several simulations.

Computation of the invariant measure for a Lévy driven SDE: Rate of convergence

“…These two assumptions imply assumptions (S a,p,q ) and (R a,p,q ) introduced in [15]. Hence, we derive the following result from [15]: Proposition 1. Let a ∈ (0, 1], p ≥ 1 and r ∈ [0, a).…”

Recursive computation of the invariant distributions of Feller processes: Revisited examples and new applications

“…Thereafter, in [14], the class of test functions for which we have lim n→∞ ν γ n f = νf a.s. (when the invariant distribution is unique) is extended to include functions with exponential growth. Finally, in [23], the results concerning the polynomial case are shown to hold for the computation of invariant measures for weakly mean reverting Levy driven diffusion processes, still using the algorithm from [12]. This extension encourages relevant perspectives concerning not only the approximation of mean reverting Brownian diffusion stationary regimes but also to treat a larger class of processes.…”

“…Our approach extends the one made in [23], and inspired by [12], for Levy processes in a weakly mean reverting setting, namely φ(y) = y a , a ∈ (0, 1] for every y ∈ [v * , ∞). Like in [23], we consider polynomial test functions, i.e. ψ p (y) = y p , with p 0 for every y ∈ [v * , ∞).…”

“…Since p 1/2, we notice that the function V p is α-Hölder for every α ∈ [2p, 1] (see Lemma 3. in [23]) and then V p is 2p-Hölder that is…”

Approximation of stationary solutions to SDEs driven by multiplicative fractional noise

“…Before going more precisely to the heart of the matter, let us mention that the numerical approximation of the stationary regime by occupation measures of Euler schemes is a classical problem in a Markov setting including diffusions and Lévy driven SDEs (see e.g. [31,20,21,22,28,29]). …”