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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
References:
[1]

P. BaldiL. Caramellino and M. G. Iovino, Pricing general barrier options: A numerical approach using sharp large deviations, Math. Finance, 9 (1999), 293-322.  doi: 10.1111/1467-9965.t01-1-00071.  Google Scholar

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I. Binder and M. Braverman, The rate of convergence of the walk on spheres algorithm, Geom. Funct. Anal., 22 (2012), 558-587.  doi: 10.1007/s00039-012-0161-z.  Google Scholar

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A. BuonocoreA. G. Nobile and L. M. Ricciardi, A new integral equation for the evaluation of first-passage-time probability densities, Adv. in Appl. Probab., 19 (1987), 784-800.  doi: 10.2307/1427102.  Google Scholar

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M. Deaconu and S. Herrmann, Simulation of hitting times for bessel processes with non-integer dimension, Bernoulli, 23 (2017), 3744-3771.  doi: 10.3150/16-BEJ866.  Google Scholar

[5]

S. Herrmann and C. Zucca, Exact simulation of the first-passage time of diffusions, J. Sci. Comput., 79 (2019), 1477-1504.  doi: 10.1007/s10915-018-00900-3.  Google Scholar

[6]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, volume 113 of Graduate Texts in Mathematics, second edition, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[7]

H. R. Lerche, Boundary Crossing of Brownian Motion, volume 40 of Lecture Notes in Statistics, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-1-4615-6569-7.  Google Scholar

[8]

M. Motoo, Some evaluations for continuous Monte Carlo method by using Brownian hitting process, Ann. Inst. Statist. Math. Tokyo, 11 (1959), 49-54.  doi: 10.1007/BF01831723.  Google Scholar

[9]

M. E. Muller, Some continuous Monte Carlo methods for the Dirichlet problem, Ann. Math. Statist., 27 (1956), 569-589.  doi: 10.1214/aoms/1177728169.  Google Scholar

[10] J. R. Norris, Markov Chains, Cambridge University Press, Cambridge, 1998.   Google Scholar
[11]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, third edition, 1999. doi: 10.1007/978-3-662-06400-9.  Google Scholar

[12]

K. K. Sabelfeld and N. A. Simonov, Random Walks on Boundary for Solving PDEs, VSP, Utrecht, 1994.  Google Scholar

[13]

K. K. Sabelfeld, Monte Carlo Methods in Boundary Value Problems, Springer Series in Computational Physics. Springer-Verlag, Berlin, 1991.  Google Scholar

[14]

L. SacerdoteO. Telve and C. Zucca, Joint densities of first hitting times of a diffusion process through two time-dependent boundaries, Adv. in Appl. Probab., 46 (2014), 186-202.  doi: 10.1239/aap/1396360109.  Google Scholar

show all references

References:
[1]

P. BaldiL. Caramellino and M. G. Iovino, Pricing general barrier options: A numerical approach using sharp large deviations, Math. Finance, 9 (1999), 293-322.  doi: 10.1111/1467-9965.t01-1-00071.  Google Scholar

[2]

I. Binder and M. Braverman, The rate of convergence of the walk on spheres algorithm, Geom. Funct. Anal., 22 (2012), 558-587.  doi: 10.1007/s00039-012-0161-z.  Google Scholar

[3]

A. BuonocoreA. G. Nobile and L. M. Ricciardi, A new integral equation for the evaluation of first-passage-time probability densities, Adv. in Appl. Probab., 19 (1987), 784-800.  doi: 10.2307/1427102.  Google Scholar

[4]

M. Deaconu and S. Herrmann, Simulation of hitting times for bessel processes with non-integer dimension, Bernoulli, 23 (2017), 3744-3771.  doi: 10.3150/16-BEJ866.  Google Scholar

[5]

S. Herrmann and C. Zucca, Exact simulation of the first-passage time of diffusions, J. Sci. Comput., 79 (2019), 1477-1504.  doi: 10.1007/s10915-018-00900-3.  Google Scholar

[6]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, volume 113 of Graduate Texts in Mathematics, second edition, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[7]

H. R. Lerche, Boundary Crossing of Brownian Motion, volume 40 of Lecture Notes in Statistics, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-1-4615-6569-7.  Google Scholar

[8]

M. Motoo, Some evaluations for continuous Monte Carlo method by using Brownian hitting process, Ann. Inst. Statist. Math. Tokyo, 11 (1959), 49-54.  doi: 10.1007/BF01831723.  Google Scholar

[9]

M. E. Muller, Some continuous Monte Carlo methods for the Dirichlet problem, Ann. Math. Statist., 27 (1956), 569-589.  doi: 10.1214/aoms/1177728169.  Google Scholar

[10] J. R. Norris, Markov Chains, Cambridge University Press, Cambridge, 1998.   Google Scholar
[11]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, third edition, 1999. doi: 10.1007/978-3-662-06400-9.  Google Scholar

[12]

K. K. Sabelfeld and N. A. Simonov, Random Walks on Boundary for Solving PDEs, VSP, Utrecht, 1994.  Google Scholar

[13]

K. K. Sabelfeld, Monte Carlo Methods in Boundary Value Problems, Springer Series in Computational Physics. Springer-Verlag, Berlin, 1991.  Google Scholar

[14]

L. SacerdoteO. Telve and C. Zucca, Joint densities of first hitting times of a diffusion process through two time-dependent boundaries, Adv. in Appl. Probab., 46 (2014), 186-202.  doi: 10.1239/aap/1396360109.  Google Scholar

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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