Processus d'Ornstein-Uhlenbeck

En mathématiques, le processus d'Ornstein-Uhlenbeck, nommé d'après Leonard Ornstein et George Uhlenbeck et aussi connu sous le nom de mean-reverting process, est un processus stochastique décrit par l'équation différentielle stochastique :dr_t = -\theta (r_t-\mu),dt + \sigma, dW_t,, où θ, μ et σ sont des paramètres déterministes et Wt est le processus de Wiener. Solution Cette équation est résolue par la méthode de variation des constantes. Appliquons le lemme d'Itō à la fonction f(r_t, t) = r_t e^{\theta t} pour obtenir : df(r_t,t) = \theta r_t e^{\theta t}, dt + e^{\theta t}, dr_t, = e^{\theta t}\theta \mu , dt + \sigma e^{\theta t}, dW_t. , En intégrant de 0 à t, on obtient : r_t e^{\theta t} = r_0 + \int_0^t e^{\theta s}\theta \mu , ds + \int_0^t \sigma e^{\theta s}, dW_s , d'où nous voyons : r_t = r_0 e^{-\theta t} + \mu(1-e^{-\theta t}) + \int_0^t \sigma e^{\theta (s-t)}, dW_s. ,

Robust Parameter Estimation for the Ornstein-Uhlenbeck Process

Sonja Rieder

In this thesis, we treat robust estimation for the parameters of the Ornstein–Uhlenbeck process, which are the mean, the variance, and the friction. We start by considering classical maximum likelihood estimation. For the simulation study, where we also investigate the choice of the time lag, we use the method of moment (MoM) estimator as initial estimator for the friction parameter of the maximum likelihood estimator (MLE). However, in several aspects the MLE is not robust. For robustification, we first derive elementary M-estimates by extending the method of M-estimation from Huber (1981). We use an intuitively robustified MoM estimate as initial estimate and compare by means of simulation the M-estimate with the MLE. This approach is, however, only ad-hoc since Huber’s minimum Fisher information and minimax asymptotic variance theory remains incomplete for simultaneous location and scale, and does not cover more general models (as for example the Ornstein–Uhlenbeck process). A more general robustness concept due to Kohl et al. (2010), Rieder (1994), and Staab (1984) is based on local asymptotic normality (LAN), asymptotically linear (AL) estimates, and shrinking neighborhoods. We then apply this concept to the Ornstein–Uhlenbeck process. As a measure of robustness, we consider the maximum asymptotic mean square error (maxasyMSE), which is determined by the influence curve (IC) of AL estimates. The IC represents the standardized influence of an individual observation on the estimator given the past. For two kind of neighborhoods (average and average square neighborhoods) we obtain optimally robust ICs. In case of average neighborhoods, their graph exhibits surprising, redescending behavior. For average square neighborhoods the graph is between the one of the elementary M-estimates and the MLE. Finally, we discuss the estimator construction, that is, the problem of constructing an estimator from the family of optimal ICs. We carry out in our context the One-Step construction dating back to LeCam and use both an intuitively robustified MoM estimate and the elementary M-estimate as initial estimate. This results in optimally AL estimates (for average and average square neighborhoods). By means of simulation we then compare the different estimators: MLE, elementary M-estimates, and optimally AL estimates. In addition, we give an application to electricity prices.

EPFL2012

Robust parameter estimation for the Ornstein-Uhlenbeck process

Sonja Rieder

In this paper, we derive elementary M- and optimally robust asymptotic linear (AL)-estimates for the parameters of an Ornstein-Uhlenbeck process. Simulation and estimation of the process are already well-studied, see Iacus (Simulation and inference for stochastic differential equations. Springer, New York, 2008). However, in order to protect against outliers and deviations from the ideal law the formulation of suitable neighborhood models and a corresponding robustification of the estimators are necessary. As a measure of robustness, we consider the maximum asymptotic mean square error (maxasyMSE), which is determined by the influence curve (IC) of AL estimates. The IC represents the standardized influence of an individual observation on the estimator given the past. In a first step, we extend the method of M-estimation from Huber (Robust statistics. Wiley, New York, 1981). In a second step, we apply the general theory based on local asymptotic normality, AL estimates, and shrinking neighborhoods due to Kohl et al. (Stat Methods Appl 19:333-354, 2010), Rieder (Robust asymptotic statistics. Springer, New York, 1994), Rieder (2003), and Staab (1984). This leads to optimally robust ICs whose graph exhibits surprising behavior. In the end, we discuss the estimator construction, i.e. the problem of constructing an estimator from the family of optimal ICs. Therefore we carry out in our context the One-Step construction dating back to LeCam (Asymptotic methods in statistical decision theory. Springer, New York, 1969) and compare it by means of simulations with MLE and M-estimator.

Springer-Verlag2012

Linking micro and macroevolution in the presence of migration

Sophie Myriam Hautphenne, Laurent Lehmann, Nicolas Salamin

Understanding macroevolutionary patterns is central to evolutionary biology. This involves the process of divergence within a species, which starts at the microevolutionary level, for instance, when two sub populations evolve towards different phenotypic optima. The speed at which these optima are reached is controlled by the degree of stabilising selection, which pushes the mean trait towards different optima in the different subpopulations, and ongoing migration that pulls the mean phenotype away from that optimum. Traditionally, macro phenotypic evolution is modelled by directional selection processes, but these models usually ignore the role of migration within species. Here, our goal is to reconcile the processes of micro and macroevolution by modelling migration as part of the speciation process. More precisely, we introduce an Ornstein-Uhlenbeck (OU) model where migration happens between two subpopulations within a branch of a phylogeny and this migration decreases over time as it happens during speciation. We then use this model to study the evolution of trait means along a phylogeny, as well as the way phenotypic disparity between species changes with successive epochs. We show that ignoring the effect of migration in sampled time-series data biases significantly the estimation of the selective forces acting upon it. We also show that migration decreases the expected phenotypic disparity between species and we analyse the effect of migration in the particular case of niche filling. We further introduce a method to jointly estimate selection and migration from time-series data. Our model extends traditional quantitative genetics results of selection and migration from a microevolutionary time frame to multiple speciation events at a macroevolutionary scale. Our results further support that not accounting for gene flow has important consequences in inferences at both the micro and macroevolutionary scale. (C) 2019 The Authors. Published by Elsevier Ltd.

ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD2020

Robust Parameter Estimation for the Ornstein-Uhlenbeck Process

Sonja Rieder

EPFL2012

Linking micro and macroevolution in the presence of migration

Sophie Myriam Hautphenne, Laurent Lehmann, Nicolas Salamin

ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD2020