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# An adaptive scheme for the approximation of dissipative systems

(Impact Factor: 1.05). 03/2005; 117(10). DOI: 10.1016/j.spa.2007.02.004
Source: arXiv

ABSTRACT 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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• "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]). "
##### Article: Approximation of stationary solutions to SDEs driven by multiplicative fractional noise
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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.
Stochastic Processes and their Applications 11/2012; 124(3). DOI:10.1016/j.spa.2013.11.004 · 1.05 Impact Factor
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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. "
##### Article: Approximation of stationary solutions of Gaussian driven Stochastic Differential Equations
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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.
Stochastic Processes and their Applications 12/2009; 121(12). DOI:10.1016/j.spa.2011.08.001 · 1.05 Impact Factor
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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. "
##### Article: Existence of the stationary regime of a Non-Markovian Stochastic Differential Equation
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ABSTRACT: In this paper, we obtain some existence results of stationary solutions to a class of SDEs driven by continuous Gaussian processes with stationary increments. We propose a constructive approach based on the study of some sequences of empirical measures of Euler schemes of these SDEs. In our main result, we obtain the functional convergence of this sequence to a stationary solution to the SDE. We also obtain some specific properties of the stationary solution. In particular, we show that, in contrast to Markovian SDEs, the initial random value of a stationary solution and the driving Gaussian process are always dependent. This emphasizes the fact that the concept of invariant distribution is definitely different to the Markovian case.