Convergence and convergence rates for approximating ergodic means of functions of solutions to stochastic differential equations with Markov switching

Mei, Hongwei; Yin, George

doi:10.1016/j.spa.2015.02.013

Cited by 11 publications

(14 citation statements)

References 11 publications

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“…The framework presented in Section 2 is well suited to this case. Our results extend the convergence results obtained in [14] and inspired by [10]. More particularly, in [14], the convergence of (ν η n ) n∈N * is established under a strongly mean reverting assumption that is φ = I d .…”

Section: The Euler Scheme For a Markov Switching Diffusionsupporting

confidence: 86%

“…In this paper, we do not restrict to that case and consider a weakly mean-reverting setting, namely φ(y) = y a , a ∈ (0, 1] for every y ∈ [v * , ∞). As a first step, we consider polynomial test functions that is ψ(y) = y p , p 1 for every y ∈ [v * , ∞) like in [14] (where p 4 is required). As a second step, still under a weakly mean-reverting setting (but where φ is not explicitly specified), we extend those results to functions ψ with exponential growth which enables to obtain convergence of the empirical measures for much wider class of test functions.…”

Section: The Euler Scheme For a Markov Switching Diffusionmentioning

confidence: 99%

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Recursive computation of invariant distributions of Feller processes

Pagès

Rey

2020

Stochastic Processes and their Applications

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This paper provides a general and abstract approach to compute invariant distributions for Feller processes. More precisely, we show that the recursive algorithm presented in [10] and based on simulation algorithms of stochastic schemes with decreasing steps can be used to build invariant measures for general Feller processes. We also propose various applications: Approximation of Markov Brownian diffusion stationary regimes with Milstein or Euler scheme and approximation of Markov switching Brownian diffusion stationary regimes using Euler scheme.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 specified explicitly), where the step sequence (γ n ) n∈N * → n→+∞ 0, a.s. weakly converges the set V, of the invariant distributions of (X t ) t 0 . To this end, we will provide as weak as possible mean reverting assumptions on the pseudo-generator of (X γ Γn ) n∈N on the one hand and appropriate rate conditions on the step sequence (γ n ) n∈N * on the other hand. Presentation of the abstract framework NotationsLet (E, |.|) be a locally compact separable metric space, we denote C(E) the set of continuous functions on E and C 0 (E) the set of continuous functions that vanish a infinity. We equip this space with the sup norm f ∞ = sup x∈E |f (x)| so that (C 0 (E), . ∞ ) is a Banach space. We will denote

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Section: The Euler Scheme For a Markov Switching Diffusionsupporting

confidence: 86%

Section: The Euler Scheme For a Markov Switching Diffusionmentioning

confidence: 99%

Recursive computation of invariant distributions of Feller processes

Pagès

Rey

2020

Stochastic Processes and their Applications

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“…See for instance [19] In this paper, we extend the abstract framework introduced in [18] in order to study rate of convergence of empirical measures (ν n ) n∈N * to ν, supposed to be unique, in the CLT. In particular we establish an abstract first order CLT (see Theorem 3.2) which enables to recover every existing results concerning rates of convergence (see [10], [12], [20] or [14]). Convergence and rate of convergence results for the Euler scheme are given as example in the end of Section 3.…”

Section: Introductionmentioning

confidence: 91%

“…Finally, in [21], 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 [10]. For a more complete overview of the studies concerning (1) for the Euler scheme, the reader can also refer to [15], [13], [20], [16], [17] or [14].…”

Section: Introductionmentioning

confidence: 96%

First and second order Central Limit Theorems for the recursive computation of the invariant distribution of a Feller process

Pagès¹,

Rey²

2018

Preprint

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This paper studies the convergence of empirical measures of a stochastic approximation toward the invariant distribution of a Feller process. In particular, we provide a general and abstract approach to establish Central Limit Theorems (CLT) with given rate . Moreover, considering weighted empirical measures of a weak order two stochastic approximation, we show its second order convergence while the CLT for standard empirical measures has order one. We also propose various applications: First order CLT for the approximation of Markov Brownian diffusion stationary regimes with Euler scheme (where we recover existing results from literature) and second order CLT for the approximation of Brownian diffusion stationary regimes using Talay [24] scheme of weak order two.

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“…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. For a more complete overview of the studies concerning (1) for the Euler scheme, the reader can also refer to [18], [15], [22], [19], [20] or [16].…”

Section: Introductionmentioning

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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Convergence and convergence rates for approximating ergodic means of functions of solutions to stochastic differential equations with Markov switching

Cited by 11 publications

References 11 publications

Recursive computation of invariant distributions of Feller processes

Recursive computation of invariant distributions of Feller processes

First and second order Central Limit Theorems for the recursive computation of the invariant distribution of a Feller process

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

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