Strong approximation of some particular one-dimensional diffusions

Madalina Deaconu (1, 2) , Samuel Herrmann (3)

1 IECL - Institut Élie Cartan de Lorraine
2 PASTA - Processus aléatoires spatio-temporels et leurs applications
3 IMB - Institut de Mathématiques de Bourgogne [Dijon]

We develop a new technique for the path approximation of one-dimensional stochastic processes. Our results apply to the Brownian motion and to some families of stochastic differential equations whose distributions could be represented as a function of a time-changed Brownian motion (usually known as L and G-classes). We are interested in the epsilon-strong approximation. We propose an explicit and easy-to-implement procedure that jointly constructs, the sequences of exit times and corresponding exit positions of some well-chosen domains. In our main results, we prove the convergence of our scheme and how to control the number of steps, which depends on the covering of a fixed time interval by intervals of random sizes. The underlying idea of our analysis is to combine results on Brownian exit times from time-depending domains (one-dimensional heat balls) and classical renewal theory. Numerical examples and issues are also developed in order to complete the theoretical results.

Mots clés: Brownian motion; linear diffusion; path simulation; Strong approximation; Strong approximation path simulation Brownian motion linear diffusion. 2010 AMS subject classifications: primary 65C05; secondary 60J60 60J65 60G17; linear diffusion. 2010 AMS subject classifications: primary 65C05; secondary 60J60; 60J65; 60G17

Domaines: Probabilités [math.PR]

Madalina Deaconu, Samuel Herrmann. Strong approximation of some particular one-dimensional diffusions. Discrete and Continuous Dynamical Systems - Series B, 2024, 29 (4), pp.1990-2017. ⟨hal-02799638v4⟩