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September  2020, 25(9): 3631-3650. doi: 10.3934/dcdsb.2020076

## Approximating exit times of continuous Markov processes

 1 Institute of Mathematics, University of Gieẞen, Arndtstr. 2, 35392 Gieẞen, Germany 2 Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany

* Corresponding author: Thomas Kruse

Received  September 2019 Revised  November 2019 Published  January 2020

Fund Project: We acknowledge the support from the German Research Foundation through the project 415705084

The time at which a one-dimensional continuous strong Markov process attains a boundary point of its state space is a discontinuous path functional and it is, therefore, unclear whether the exit time can be approximated by hitting times of approximations of the process. We prove a functional limit theorem for approximating weakly both the paths of the Markov process and its exit times. In contrast to the functional limit theorem in [3] for approximating the paths, we impose a stronger assumption here. This is essential, as we present an example showing that the theorem extended with the convergence of the exit times does not hold under the assumption in [3]. However, the EMCEL scheme introduced in [3] satisfies the assumption of our theorem, and hence we have a scheme capable of approximating both the process and its exit times for every one-dimensional continuous strong Markov process, even with irregular behavior (e.g., a solution of an SDE with irregular coefficients or a Markov process with sticky features). Moreover, our main result can be used to check for some other schemes whether the exit times converge. As an application we verify that the weak Euler scheme is capable of approximating the absorption time of the CEV diffusion and that the scale-transformed weak Euler scheme for a squared Bessel process is capable of approximating the time when the squared Bessel process hits zero.

Citation: Thomas Kruse, Mikhail Urusov. Approximating exit times of continuous Markov processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3631-3650. doi: 10.3934/dcdsb.2020076
##### References:
 [1] V. M. Abramov, F. C. Klebaner and R. S. Liptser, The Euler-Maruyama approximations for the CEV model, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 1-14.  doi: 10.3934/dcdsb.2011.16.1.  Google Scholar [2] S. Ankirchner, T. Kruse and M. Urusov, Numerical approximation of irregular SDEs via Skorokhod embeddings, J. Math. Anal. Appl., 440 (2016), 692-715.  doi: 10.1016/j.jmaa.2016.03.055.  Google Scholar [3] S. Ankirchner, T. Kruse and M. Urusov, A functional limit theorem for coin tossing Markov chains, Preprint, arXiv: 1902.06249. Google Scholar [4] S. Ankirchner, T. Kruse and M. Urusov, Wasserstein convergence rates for random bit approximations of continuous markov processes, Preprint, arXiv: 1903.07880. Google Scholar [5] R. F. Bass, A stochastic differential equation with a sticky point, Electron. J. Probab., 19 (2014), 22pp. doi: 10.1214/EJP.v19-2350.  Google Scholar [6] A. Beskos, O. Papaspiliopoulos and G. O. Roberts, Retrospective exact simulation of diffusion sample paths with applications, Bernoulli, 12 (2006), 1077-1098.   Google Scholar [7] B. Bouchard, S. Geiss and E. Gobet, First time to exit of a continuous Itȏ process: General moment estimates and $L_1$-convergence rate for discrete time approximations, Bernoulli, 23 (2017), 1631-1662.  doi: 10.3150/15-BEJ791.  Google Scholar [8] L. Breiman, Probability, vol. 7 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992, Corrected reprint of the 1968 original. doi: 10.1137/1.9781611971286.  Google Scholar [9] C. Bruggeman and J. Ruf, A one-dimensional diffusion hits points fast, Electron. Commun. Probab., 21 (2016), Paper No. 22, 7pp. doi: 10.1214/16-ECP4544.  Google Scholar [10] P. Chigansky and F. C. Klebaner, The Euler-Maruyama approximation for the absorption time of the CEV diffusion, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1455-1471.  doi: 10.3934/dcdsb.2012.17.1455.  Google Scholar [11] M. Deaconu and S. Herrmann, Hitting time for Bessel processes—walk on moving spheres algorithm (WoMS), Ann. Appl. Probab., 23 (2013), 2259-2289.  doi: 10.1214/12-AAP900.  Google Scholar [12] 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 [13] H.-J. Engelbert and G. Peskir, Stochastic differential equations for sticky Brownian motion, Stochastics, 86 (2014), 993-1021.  doi: 10.1080/17442508.2014.899600.  Google Scholar [14] H.-J. Engelbert and W. Schmidt, On solutions of one-dimensional stochastic differential equations without drift, Z. Wahrsch. Verw. Gebiete, 68 (1985), 287-314.  doi: 10.1007/BF00532642.  Google Scholar [15] S. N. Ethier, Limit theorems for absorption times of genetic models, Ann. Probab., 7 (1979), 622–638, http://links.jstor.org/sici?sici=0091-1798(197908)7:4<622:LTFATO>2.0.CO;2-Q&origin=MSN. doi: 10.1214/aop/1176994986.  Google Scholar [16] I. Gyöngy, A note on Euler's approximations, Potential Analysis, 8 (1998), 205-216.  doi: 10.1023/A:1008605221617.  Google Scholar [17] H. Hajri, M. Caglar and M. Arnaudon, Application of stochastic flows to the sticky Brownian motion equation, Electron. Commun. Probab., 22 (2017), Paper No. 3, 10pp. doi: 10.1214/16-ECP37.  Google Scholar [18] S. Herrmann and C. Zucca, Exact simulation of the first-passage time of diffusions, J. Sci. Comput., 79 (2019), 1477–1504, arXiv: 1705.06881v1. doi: 10.1007/s10915-018-00900-3.  Google Scholar [19] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, vol. 113 of Graduate Texts in Mathematics, 2nd edition, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar [20] I. Karatzas and J. Ruf, Distribution of the time to explosion for one-dimensional diffusions, Probab. Theory Related Fields, 164 (2016), 1027-1069.  doi: 10.1007/s00440-015-0625-9.  Google Scholar [21] I. Karatzas, A. N. Shiryaev and M. Shkolnikov, On the one-sided Tanaka equation with drift, Electron. Commun. Probab., 16 (2011), 664-677.  doi: 10.1214/ECP.v16-1665.  Google Scholar [22] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, vol. 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edition, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-06400-9.  Google Scholar [23] L. C. G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales. Vol. 2, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2000, Itô calculus, Reprint of the second (1994) edition. doi: 10.1017/CBO9781107590120.  Google Scholar [24] D. Taguchi and A. Tanaka, On the Euler–Maruyama scheme for degenerate stochastic differential equations with non-sticky boundary condition, Preprint, arXiv: 1902.05712. Google Scholar [25] L. Yan, The Euler scheme with irregular coefficients, The Annals of Probability, 30 (2002), 1172-1194.  doi: 10.1214/aop/1029867124.  Google Scholar

show all references

##### References:
 [1] V. M. Abramov, F. C. Klebaner and R. S. Liptser, The Euler-Maruyama approximations for the CEV model, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 1-14.  doi: 10.3934/dcdsb.2011.16.1.  Google Scholar [2] S. Ankirchner, T. Kruse and M. Urusov, Numerical approximation of irregular SDEs via Skorokhod embeddings, J. Math. Anal. Appl., 440 (2016), 692-715.  doi: 10.1016/j.jmaa.2016.03.055.  Google Scholar [3] S. Ankirchner, T. Kruse and M. Urusov, A functional limit theorem for coin tossing Markov chains, Preprint, arXiv: 1902.06249. Google Scholar [4] S. Ankirchner, T. Kruse and M. Urusov, Wasserstein convergence rates for random bit approximations of continuous markov processes, Preprint, arXiv: 1903.07880. Google Scholar [5] R. F. Bass, A stochastic differential equation with a sticky point, Electron. J. Probab., 19 (2014), 22pp. doi: 10.1214/EJP.v19-2350.  Google Scholar [6] A. Beskos, O. Papaspiliopoulos and G. O. Roberts, Retrospective exact simulation of diffusion sample paths with applications, Bernoulli, 12 (2006), 1077-1098.   Google Scholar [7] B. Bouchard, S. Geiss and E. Gobet, First time to exit of a continuous Itȏ process: General moment estimates and $L_1$-convergence rate for discrete time approximations, Bernoulli, 23 (2017), 1631-1662.  doi: 10.3150/15-BEJ791.  Google Scholar [8] L. Breiman, Probability, vol. 7 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992, Corrected reprint of the 1968 original. doi: 10.1137/1.9781611971286.  Google Scholar [9] C. Bruggeman and J. Ruf, A one-dimensional diffusion hits points fast, Electron. Commun. Probab., 21 (2016), Paper No. 22, 7pp. doi: 10.1214/16-ECP4544.  Google Scholar [10] P. Chigansky and F. C. Klebaner, The Euler-Maruyama approximation for the absorption time of the CEV diffusion, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1455-1471.  doi: 10.3934/dcdsb.2012.17.1455.  Google Scholar [11] M. Deaconu and S. Herrmann, Hitting time for Bessel processes—walk on moving spheres algorithm (WoMS), Ann. Appl. Probab., 23 (2013), 2259-2289.  doi: 10.1214/12-AAP900.  Google Scholar [12] 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 [13] H.-J. Engelbert and G. Peskir, Stochastic differential equations for sticky Brownian motion, Stochastics, 86 (2014), 993-1021.  doi: 10.1080/17442508.2014.899600.  Google Scholar [14] H.-J. Engelbert and W. Schmidt, On solutions of one-dimensional stochastic differential equations without drift, Z. Wahrsch. Verw. Gebiete, 68 (1985), 287-314.  doi: 10.1007/BF00532642.  Google Scholar [15] S. N. Ethier, Limit theorems for absorption times of genetic models, Ann. Probab., 7 (1979), 622–638, http://links.jstor.org/sici?sici=0091-1798(197908)7:4<622:LTFATO>2.0.CO;2-Q&origin=MSN. doi: 10.1214/aop/1176994986.  Google Scholar [16] I. Gyöngy, A note on Euler's approximations, Potential Analysis, 8 (1998), 205-216.  doi: 10.1023/A:1008605221617.  Google Scholar [17] H. Hajri, M. Caglar and M. Arnaudon, Application of stochastic flows to the sticky Brownian motion equation, Electron. Commun. Probab., 22 (2017), Paper No. 3, 10pp. doi: 10.1214/16-ECP37.  Google Scholar [18] S. Herrmann and C. Zucca, Exact simulation of the first-passage time of diffusions, J. Sci. Comput., 79 (2019), 1477–1504, arXiv: 1705.06881v1. doi: 10.1007/s10915-018-00900-3.  Google Scholar [19] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, vol. 113 of Graduate Texts in Mathematics, 2nd edition, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar [20] I. Karatzas and J. Ruf, Distribution of the time to explosion for one-dimensional diffusions, Probab. Theory Related Fields, 164 (2016), 1027-1069.  doi: 10.1007/s00440-015-0625-9.  Google Scholar [21] I. Karatzas, A. N. Shiryaev and M. Shkolnikov, On the one-sided Tanaka equation with drift, Electron. Commun. Probab., 16 (2011), 664-677.  doi: 10.1214/ECP.v16-1665.  Google Scholar [22] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, vol. 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edition, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-06400-9.  Google Scholar [23] L. C. G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales. Vol. 2, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2000, Itô calculus, Reprint of the second (1994) edition. doi: 10.1017/CBO9781107590120.  Google Scholar [24] D. Taguchi and A. Tanaka, On the Euler–Maruyama scheme for degenerate stochastic differential equations with non-sticky boundary condition, Preprint, arXiv: 1902.05712. Google Scholar [25] L. Yan, The Euler scheme with irregular coefficients, The Annals of Probability, 30 (2002), 1172-1194.  doi: 10.1214/aop/1029867124.  Google Scholar
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