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August  2020, 25(8): 3199-3215. doi: 10.3934/dcdsb.2020058

Exit problem for Ornstein-Uhlenbeck processes: A random walk approach

 Institut de Mathématiques de Bourgogne - UMR 5584, CNRS, Université de Bourgogne Franche-Comté, F-21000 Dijon, France

* Corresponding author: Samuel Herrmann

Received  June 2019 Revised  October 2019 Published  February 2020

In order to approximate the exit time of a one-dimensional diffusion process, we propose an algorithm based on a random walk. Such an algorithm so-called Walk on Moving Spheres was already introduced in the Brownian context. The aim is therefore to generalize this numerical approach to the Ornstein-Uhlenbeck process and to describe the efficiency of the method.

Citation: Samuel Herrmann, Nicolas Massin. Exit problem for Ornstein-Uhlenbeck processes: A random walk approach. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3199-3215. doi: 10.3934/dcdsb.2020058
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A sample of the algorithm for the O.U. exit time with parameters $\theta = 0.1$ and $\sigma = 1$. We observe the walk on spheres associated with the diffusion process starting at $x = 5$ and moving in the interval $[2,7]$. The algorithm corresponding to $\epsilon = 0,5$ is represented by the plain style spheroids whereas the case $\epsilon = 10^{-3}$ corresponds to the whole sequence of spheroids. In both cases we set $\gamma = 10^{-6}$
Histogram of the outcome variable for the O.U. with parameters $\theta = 0.1$ and $\sigma = 1$ when the stopped diffusion process starts at 5 and involves in the interval [2, 7] with $\epsilon = 10^{-3}$ and $\gamma = 10^{-6}$
Histogram of the approximated first exit time of the interval $[a,b]$ using the WOMS algorithm and approximated p.d.f. of the first passage time through the level $b$ (curve). Here $X_0 = -3$, $\theta = 1$, $\sigma = 1$ and $[a,b] = [-10,-1]$
Simulation of the O.U. exit time from the interval $[2,7]$. The starting position is $X_0 = 5$ and the parameters are given by $\theta = 0.1$, $\sigma = 1$ and $\gamma = 10^{-6}$. Histogram of the number of steps observed for $\epsilon = 10^{-3}$
Simulation of the O.U. exit time from the interval $[2,7]$. The starting position is $X_0 = 5$ and the parameters are given by $\theta = 0.1$, $\sigma = 1$ and $\gamma = 10^{-6}$. Average number of steps versus $\epsilon$ (in logarithmic scale)
Error bound $\Xi$ versus $\epsilon$ for different values of $\theta$ with $\sigma = 1$, $a = -1$, $b = 1$, $\gamma = 1$
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