Ergodic approximation of the distribution of a stationary diffusion: Rate of convergence

Abstract

We extend to Lipschitz continuous functionals either of the true paths or of the Euler scheme with decreasing step of a wide class of Brownian ergodic diffusions, the Central Limit Theorems formally established for their marginal empirical measure of these processes (which is classical for the diffusions and more recent as concerns their discretization schemes). We illustrate our results by simulations in connection with barrier option pricing.

A mixed-step algorithm for the approximation of the stationary regime of a diffusion

“…However in some particular cases (fortunately often those of interest) the functional F can be computed recursively (see [18] for details) so that our estimate κ F × m should be replaced by κ F × c 0 which makes the resulting global complexity comparable to our new approach. As a conclusion, our new procedure yields a better control of the complexity for all (computable) functionals F .…”

“…We derive from Lemma 3.2(i) of [18] and from the polynomial growth of Φ g that there exists r > 0 such that sup…”

“…In a series of papers ( [11,13,19,20,17,18]) going back to [10], have been investigated the properties of an Euler scheme with decreasing step as a tool for the numerical approximation of the [steady/stationary] regime of a diffusion, possibly with jumps, satisfying some [stability/mean reverting] conditions. The purpose of the present paper is to propose and investigate a variant of the original procedure sharing the similar properties in terms of convergence and rate but with a lower complexity, especially in its functional form, i.e.…”

“…Basically when the step goes to 0 fast enough the empirical measure behaves like the empirical measure of the diffusion itself and satisfies the standard CLT (see e.g. [2] and [18] for results on the diffusion itself). When the convergence of (γ n ) to 0 is too slow, the whole procedure is slowed down and satisfies an a.s. (or at least in probability) rate of convergence property.…”

“…These empirical measures on path spaces have in turn been extensively investigated in [17,18]. However, their practical implementation suffers in practice from a significant drawback: one cannot prevent the complexity of the computation of F (X (Γn),T ) to go to infinity as n grows since more and more iterations of the Euler schemes are needed to "cover" a laps of time equal to T .…”