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1

Limit behavior of the invariant measure for Langevin~dynamics

Barrera Gerardo

Probability and Mathematical Statistics

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2022

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Vol. 42, Fasc. 1

143--162

EN

We consider the Langevin dynamics on Rd with an overdamped vector field and driven by multiplicative Brownian noise of small amplitude √ϵ, ϵ>0. Under suitable assumptions on the vector field and the diffusion coefficient, it is well-known that it has a unique invariant probability measure μ ϵ . We prove that as ε tends to zero, the probability measure ϵd/2μ ϵ(√ϵdx) converges in the p--Wasserstein distance for p∈[1,2] to a Gaussian measure with zero-mean vector and non-degenerate covariance matrix which solves a Lyapunov matrix equation. Moreover, the error term is estimated. We emphasize that generically no explicit formula for μϵ can be found.

2

Fractional lower order covariance based-estimator for Ornstein-Uhlenbeck process with stable distribution

Kruczek Piotr, Żuławiński Wojciech, Pagacz Patrycja, Wyłomańska Agnieszka

Mathematica Applicanda

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2019

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Vol. 47, no. 2

269--292

EN

The Ornstein-Uhlenbeck model is one of the most popular stochastic processes. It has found many interesting applications including physical phenomena. However, for many real data, the classical Ornstein-Uhlenbeck process cannot be applied. It is related to the fact that for many phenomena the vectors of observations exhibit the so-called heavy-tailed behaviour. In such cases, the modifications of the classical models need to be used. In this paper, we analyze the Ornstein-Uhlenbeck process based on stable distribution. This distribution is one of the most classical members of the heavy-tailed class of distributions. In the literature, one can find various applications of stable processes. However, the heavy-tailed property implies that the classical methods of estimation and statistical investigation cannot be applied. In this paper, we propose a new method of estimation of the stable Ornstein-Uhlenbeck process. This technique is based on the alternative measure of dependence, called fractional lower order covariance, which replaces the classical covariance for infinite-variance distribution. The proposed research is a continuation of the authors' previous studies, where the measure called covariation was proposed as the base for the estimation technique. We introduce the stable Ornstein-Uhlenbeck process and remind its main properties. In the main part, we define the new estymator of the parameters for discrete representation of the Ornstein-Uhlenbeck process. Its effectiveness is checked by Monte Carlo simulations.

PL

Proces Ornsteina-Uhlenbecka jest jednym z najbardziej popularnych procesów stochastycznych. Znalazł on wiele ciekawych praktycznych zastosowań. Należy jednak zwrócić uwagę, że klasyczny proces Ornsteina-Uhlenbecka nie może być zastosowany dla wielu danych rzeczywistych, ponieważ często pochodzą one z rozkładów ciężko- ogonwych, dla których nie istnieje drugi moment. W takich przypadkach niezbędna jest modyfikacja klasycznego modelu z wykorzystaniem rozkładu stabilnego. Z powodu zastosowania rozkładu stabilnego niezbędne jest użycie innej metody estymacji niż bazującej na autokowariancji. Zaproponowana została nowa metoda bazująca na ułamkowych momentach. Praca jest kontynuacją wcześniej otrzymanych rezultatów dla innej alternatywnej miary zależności, kowariacji. W pracy przypomniana została definicja stabilnego procesu Ornsteina-Uhlenbecka wraz z propozycją nowych estymatorów dla parametrów tego procesu.W celu sprawdzenia ich właściwości wykonane zostały symulacje Monte Carlo.

3

Diffusion approximation of recurrent schemes for financial markets, with application to the Ornstein-Uhlenbeck process

Mishura Y.

Opuscula Mathematica

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2015

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Vol. 35, no. 1

99--116

EN

We adapt the general conditions of the weak convergence for the sequence of processes with discrete time to the diffusion process towards the weak convergence for the discrete-time models of a financial market to the continuous-time diffusion model. These results generalize a classical scheme of the weak convergence for discrete-time markets to the Black-Scholes model. We give an explicit and direct method of approximation by a recurrent scheme. As an example, an Ornstein-Uhlenbeck process is considered as a limit model.

4

Lévy processes and stochastic integrals in Banach spaces

Applebaum D.

Probability and Mathematical Statistics

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2007

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Vol. 27, Fasc. 1

75--88

EN

We review infinite divisibility and Lévy processes in Banach spaces and discuss the relationship with notions of type and cotype. The Lévy-Itô decomposition is described. Strong, weak and Pettis-style notions of stochastic integral are introduced and applied to construct generalised Ornstein-Uhlenbeck processes