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

  • * Corresponding author: Samuel Herrmann

    * Corresponding author: Samuel Herrmann 
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  • 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:

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  • 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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