An adaptive scheme for the approximation of dissipative systems

Stochastic Processes and their Applications (Impact Factor: 1.06). 03/2005; 117(10). DOI: 10.1016/
Source: arXiv


We propose a new scheme for the long time approximation of a diffusion when the drift vector field is not globally Lipschitz. Under this assumption, regular explicit Euler scheme --with constant or decreasing step-- may explode and implicit Euler scheme are CPU-time expensive. The algorithm we introduce is explicit and we prove that any weak limit of the weighted empirical measures of this scheme is a stationary distribution of the stochastic differential equation. Several examples are presented including gradient dissipative systems and Hamiltonian dissipative systems.

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    • "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. when trying to compute the expectation of a functional of the process (over a finite time interval [0, T ]) with respect to the stationary distribution of the process. "
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    ABSTRACT: In some recent papers, some procedures based on some weighted empirical measures related to decreasing-step Euler schemes have been investigated to approximate the stationary regime of a diffusion (possibly with jumps) for a class of functionals of the process. This method is efficient but needs the computation of the function at each step. To reduce the complexity of the procedure (especially for functionals), we propose in this paper to study a new scheme, called the mixed-step scheme, where we only keep some regularly time-spaced values of the Euler scheme. Our main result is that, when the coefficients of the diffusion are smooth enough, this alternative does not change the order of the rate of convergence of the procedure. We also investigate a Richardson–Romberg method to speed up the convergence and show that the variance of the original algorithm can be preserved under a uniqueness assumption for the invariant distribution of the “duplicated” diffusion, condition which is extensively discussed in the paper. Finally, we conclude by giving sufficient “asymptotic confluence” conditions for the existence of a smooth solution to a discrete version of the associated Poisson equation, condition which is required to ensure the rate of convergence results.
    Full-text · Article · Jan 2014 · Stochastic Processes and their Applications
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    • "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]). "
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    ABSTRACT: In a previous paper, we studied the ergodic properties of an Euler scheme of a stochastic differential equation with a Gaussian additive noise in order to approximate the stationary regime of such equation. We now consider the case of multiplicative noise when the Gaussian process is a fractional Brownian Motion with Hurst parameter H>1/2 and obtain some (functional) convergences properties of some empirical measures of the Euler scheme to the stationary solutions of such SDEs.
    Full-text · Article · Nov 2012 · Stochastic Processes and their Applications
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    • "Denoting these stationary solutions by Y (∞,γ) , we show in a second step, that (Y (∞,γ) ) γ is tight for the uniform convergence on compact sets and that its weak limits (when γ → 0) are stationary solutions to (1). For a Markovian SDE, this type of approach is used as a way of numerical approximation of the invariant distribution and more generally of the distribution of the Markov process when stationary (see [17], [9], [10], [12], [16], [15]). Here, even if the discrete model can be simulated, we essentially use it as a natural way of construction of stationary solutions of the continuous model and the computation problems are out of the scope of this paper. "
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    ABSTRACT: We study sequences of empirical measures of Euler schemes associated to some non-Markovian SDEs: SDEs driven by Gaussian processes with stationary increments. We obtain the functional convergence of this sequence to a stationary solution to the SDE. Then, we end the paper by some specific properties of this stationary solution. We show that, in contrast to Markovian SDEs, its initial random value and the driving Gaussian process are always dependent. However, under an integral representation assumption, we also obtain that the past of the solution is independent to the future of the underlying innovation process of the Gaussian driving process.
    Preview · Article · Dec 2009 · Stochastic Processes and their Applications
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