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

• * Corresponding author: Samuel Herrmann
• 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.

Mathematics Subject Classification: Primary: 65C05; Secondary: 60J60, 60G40, 60G46.

 Citation: • • Figure 1.  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 $\gamma = 10^{-6}$

Figure 2.  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 $\gamma = 10^{-6}$

Figure 3.  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 $[a,b] = [-10,-1]$

Figure 4.  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 $\epsilon = 10^{-3}$

Figure 5.  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)

Figure 6.  Error bound $\Xi$ versus $\epsilon$ for different values of $\theta$ with $\sigma = 1$, $a = -1$, $b = 1$, $\gamma = 1$

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