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

In the last decade, there has been a growing interest to use Wishart processes for modelling, especially for financial applications. However, there are still few studies on the estimation of its parameters. Here, we study the Maximum Likelihood Estimator (MLE) in order to estimate the drift parameters of a Wishart process. We obtain precise convergence rates and limits for this estimator in the ergodic case and in some nonergodic cases. We check that the MLE achieves the optimal convergence rate in each case. Motivated by this study, we also present new results on the Laplace transform that extend the recent findings of Gnoatto and Grasselli [17] and are of independent interest.

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The literature about non-linear dynamics offers a few recommendations, which sometimes are divergent, about the criteria to be used in order to select the optimal calculus parameters in the estimation of Lyapunov exponents by direct methods. These few recommendations are circumscribed to the analysis of chaotic systems. We have found no recommendation for the estimation of A starting from the time series of classic systems. The reason for this is the interest in distinguishing variability due to a chaotic behavior of determinist dynamic systems of variability caused by white noise or linear stochastic processes, and less in the identification of non-linear terms from the analysis of time series.In this study we have centered in the dependence of the Lyapunov exponent, obtained by means of direct estimation, of the initial distance and the time evolution. We have used generated series of chaotic systems and generated series of classic systems with varying complexity. To generate the series we have used the logistic map.

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

In single photon emission computed tomography (SPECT), attenuation and scatter introduce important artefacts in the reconstructed images leading to mis diagnosis for patient's follow-up. Furthermore, by using Monte Carlo simulation (MCS), physical effects undergone by photons during the SPECT exam can be precisely modeled and accounted for during iterative reconstruction, which improves the quality of the image. However, MCS are time consuming and therefore inappropriate for the rate of daily exams performed in clinical routine. Our work is based on the assumption that patients are composed of identical biological tissues and that photon propagation in an element volume of a given tissue is similar and reproducible from one subject to another. We hence propose to accelerate modeling of the physical effects occurring in emission tomography making it adequate for daily exam by using the approach of scatter pre-calculated database. The developed efficient patientdependent attenuation and scatter correction were implemented on a GPU architecture on a state-ofart single-processor workstation and yielded to a speed-up factor in time computing of four orders of magnitude. Results presented in this proof of concept study are in good agreement with full MCS.