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

Panloup, Fabien

doi:10.1016/j.spa.2007.09.007

Cited by 16 publications

(47 citation statements)

References 16 publications

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“…Let us also mention the work of Panloup [Pan08b], where under similar assumptions for stochastic equation driven by a Lévy process, the convergence of the decreasing time step algorithm towards the invariant measure of the stochastic process is established (see also [Pan08a] for the CLT 2. With our tensor notations,…”

Section: Hypotheses (C1)mentioning

confidence: 99%

Sharp non-asymptotic concentration inequalities for the approximation of the invariant distribution of a diffusion

Honoré

2020

Stochastic Processes and their Applications

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For an ergodic Brownian diffusion with invariant measure ν, we consider a sequence of empirical distributions (νn) n≥1 associated with an approximation scheme with decreasing time step (γn) n≥1 along an adapted regular enough class of test functions f such that f −ν(f ) is a coboundary of the infinitesimal generator A. Denote by σ the diffusion coefficient and ϕ the solution of the Poisson equation Aϕ = f − ν(f ). When the square norm |σ * ∇ϕ| 2 lies in the same coboundary class as f , we establish sharp non-asymptotic concentration bounds for suitable normalizations of νn(f ) − ν(f ). Our bounds are optimal in the sense that they match the asymptotic limit obtained by Lamberton and Pagès in [LP02], for a certain large deviation regime. In particular, this allows us to derive sharp non-asymptotic confidence intervals. Eventually, we are able to handle, up to an additional constraint on the time steps, Lipschitz sources f in an appropriate non-degenerate setting.

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Section: Hypotheses (C1)mentioning

confidence: 99%

Sharp non-asymptotic concentration inequalities for the approximation of the invariant distribution of a diffusion

Honoré

2020

Stochastic Processes and their Applications

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“…y ∈ R + . Assume that (22) and B(φ) (see (24)) hold, that ρ < s, and that ∀λ λ, ∃C 0, ∀n ∈ N, E[exp(λV p (X Γ n+1 ))|X Γn ] C exp(λV p (X Γn )).…”

Section: Proof Of Growth Control and Step Weight Assumptionsmentioning

confidence: 99%

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

Pagès¹,

Rey

2019

Monte Carlo Methods and Applications

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In this paper, we show that the abstract framework developed in [21] and inspired by [12] can be used to build invariant distributions for Brownian diffusion processes using the Milstein scheme and for diffusion processes with censored jump using the Euler scheme. Both studies rely on a weakly mean reverting setting for both cases. For the Milstein scheme we prove the convergence for test functions with polynomial (Wasserstein convergence) and exponential growth. For the Euler scheme of diffusion processes with censored jump we prove the convergence for test functions with polynomial growth.In this section, we show that the empirical measures defined in the same way as in (1) and built from an approximation (X γ Γn ) n∈N of a Feller process (X t ) t 0 (which are not explicitly specified), where

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“…When the Poisson equation related to the infinitesimal generator has a solution, this convergence is ruled by a Central Limit Theorem (CLT ): this has been extensively investigated in the literature (for continuous Markov processes, see [4], for the Euler scheme with decreasing step of Brownian diffusions, see [18,21]). As concerns Lévy driven SDEs, see [24]. Our aim in this paper is to extend some of these rate results to functionals of the path process and its associated Euler scheme with decreasing step, i.e.…”

Section: Introductionmentioning

confidence: 96%

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

Pagès¹,

Panloup²

2012

Ann. Appl. Probab.

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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.

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

Cited by 16 publications

References 16 publications

Sharp non-asymptotic concentration inequalities for the approximation of the invariant distribution of a diffusion

Sharp non-asymptotic concentration inequalities for the approximation of the invariant distribution of a diffusion

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

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

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