• Previous Article
    Stability of delay differential equations with fading stochastic perturbations of the type of white noise and poisson's jumps
  • DCDS-B Home
  • This Issue
  • Next Article
    Influence of mutations in phenotypically-structured populations in time periodic environment
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. AbramovF. 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. AnkirchnerT. 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. BeskosO. Papaspiliopoulos and G. O. Roberts, Retrospective exact simulation of diffusion sample paths with applications, Bernoulli, 12 (2006), 1077-1098.   Google Scholar

[7]

B. BouchardS. 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. KaratzasA. 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. AbramovF. 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. AnkirchnerT. 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. BeskosO. Papaspiliopoulos and G. O. Roberts, Retrospective exact simulation of diffusion sample paths with applications, Bernoulli, 12 (2006), 1077-1098.   Google Scholar

[7]

B. BouchardS. 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. KaratzasA. 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

[1]

Fernando J. Sánchez-Salas. Dimension of Markov towers for non uniformly expanding one-dimensional systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1447-1464. doi: 10.3934/dcds.2003.9.1447

[2]

Mathias Staudigl. A limit theorem for Markov decision processes. Journal of Dynamics & Games, 2014, 1 (4) : 639-659. doi: 10.3934/jdg.2014.1.639

[3]

Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005

[4]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037

[5]

Xi Zhu, Meixia Li, Chunfa Li. Consensus in discrete-time multi-agent systems with uncertain topologies and random delays governed by a Markov chain. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020111

[6]

Ralf Banisch, Carsten Hartmann. A sparse Markov chain approximation of LQ-type stochastic control problems. Mathematical Control & Related Fields, 2016, 6 (3) : 363-389. doi: 10.3934/mcrf.2016007

[7]

Kun Fan, Yang Shen, Tak Kuen Siu, Rongming Wang. On a Markov chain approximation method for option pricing with regime switching. Journal of Industrial & Management Optimization, 2016, 12 (2) : 529-541. doi: 10.3934/jimo.2016.12.529

[8]

Angelica Pachon, Federico Polito, Costantino Ricciuti. On discrete-time semi-Markov processes. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020170

[9]

Pavel Chigansky, Fima C. Klebaner. The Euler-Maruyama approximation for the absorption time of the CEV diffusion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1455-1471. doi: 10.3934/dcdsb.2012.17.1455

[10]

Shi Jin, Min Tang, Houde Han. A uniformly second order numerical method for the one-dimensional discrete-ordinate transport equation and its diffusion limit with interface. Networks & Heterogeneous Media, 2009, 4 (1) : 35-65. doi: 10.3934/nhm.2009.4.35

[11]

Samuel N. Cohen, Lukasz Szpruch. On Markovian solutions to Markov Chain BSDEs. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 257-269. doi: 10.3934/naco.2012.2.257

[12]

Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831

[13]

Peter Giesl, James McMichen. Determination of the area of exponential attraction in one-dimensional finite-time systems using meshless collocation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1835-1850. doi: 10.3934/dcdsb.2018094

[14]

Richard Sharp. Conformal Markov systems, Patterson-Sullivan measure on limit sets and spectral triples. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2711-2727. doi: 10.3934/dcds.2016.36.2711

[15]

Ralf Banisch, Carsten Hartmann. Addendum to "A sparse Markov chain approximation of LQ-type stochastic control problems". Mathematical Control & Related Fields, 2017, 7 (4) : 623-623. doi: 10.3934/mcrf.2017023

[16]

Zhi-An Wang, Kun Zhao. Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model. Communications on Pure & Applied Analysis, 2013, 12 (6) : 3027-3046. doi: 10.3934/cpaa.2013.12.3027

[17]

Maria do Carmo Pacheco de Toledo, Sergio Muniz Oliva. A discretization scheme for an one-dimensional reaction-diffusion equation with delay and its dynamics. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1041-1060. doi: 10.3934/dcds.2009.23.1041

[18]

Benjamin Jourdain, Julien Reygner. Optimal convergence rate of the multitype sticky particle approximation of one-dimensional diagonal hyperbolic systems with monotonic initial data. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4963-4996. doi: 10.3934/dcds.2016015

[19]

Denis Mercier, Serge Nicaise. Existence results for general systems of differential equations on one-dimensional networks and prewavelets approximation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 273-300. doi: 10.3934/dcds.1998.4.273

[20]

Toyohiko Aiki, Martijn Anthonissen, Adrian Muntean. On a one-dimensional shape-memory alloy model in its fast-temperature-activation limit. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 15-28. doi: 10.3934/dcdss.2012.5.15

2019 Impact Factor: 1.27

Article outline

[Back to Top]