The simulation of exit times for diffusion processes is a challenging task since it concerns many applications in different fields like mathematical finance, neuroscience, reliability… The usual procedure is to use discretization schemes which unfortunately introduce some error in the target distribution. Our aim is to present a new algorithm which simulates exactly the exit time for one-dimensional diffusions. This acceptance-rejection algorithm requires to simulate exactly the exit time of the Brownian motion on one side and the Brownian position at a given time, constrained not to have exit before, on the other side. Crucial tools in this study are the Girsanov transformation, the convergent series method for the simulation of random variables and the classical rejection sampling. The efficiency of the method is described through theoretical results and numerical examples.
Accepté le :
Publié le :
DOI : 10.1051/m2an/2019077
Mots-clés : Exit time, Brownian motion, diffusion processes, Girsanov’s transformation, rejection sampling, exact simulation, randomized algorithm, conditioned Brownian motion
@article{M2AN_2020__54_3_811_0,
author = {Herrmann, Samuel and Zucca, Cristina},
title = {Exact simulation of first exit times for one-dimensional diffusion processes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {811--844},
publisher = {EDP-Sciences},
volume = {54},
number = {3},
year = {2020},
doi = {10.1051/m2an/2019077},
mrnumber = {4080787},
zbl = {07197800},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2019077/}
}
TY - JOUR AU - Herrmann, Samuel AU - Zucca, Cristina TI - Exact simulation of first exit times for one-dimensional diffusion processes JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2020 SP - 811 EP - 844 VL - 54 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2019077/ DO - 10.1051/m2an/2019077 LA - en ID - M2AN_2020__54_3_811_0 ER -
%0 Journal Article %A Herrmann, Samuel %A Zucca, Cristina %T Exact simulation of first exit times for one-dimensional diffusion processes %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2020 %P 811-844 %V 54 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2019077/ %R 10.1051/m2an/2019077 %G en %F M2AN_2020__54_3_811_0
Herrmann, Samuel; Zucca, Cristina. Exact simulation of first exit times for one-dimensional diffusion processes. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 3, pp. 811-844. doi : 10.1051/m2an/2019077. http://www.numdam.org/articles/10.1051/m2an/2019077/
, and , Representations of the first hitting time density of an Ornstein-Uhlenbeck process. Stoch. Models 21 (2005) 967–980. | DOI | MR | Zbl
, and , Pricing general barrier options: a numerical approach using sharp large deviations. Math. Finance 9 (1999) 293–322. | DOI | MR | Zbl
, Diffusions and elliptic operators. In: Probability and its Applications (New York), Springer-Verlag, New York, 1998. | MR | Zbl
and , Exact simulation of diffusions. Ann. Appl. Probab. 15 (2005) 2422–2444. | DOI | MR | Zbl
, and , Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12 (2006) 1077–1098. | DOI | MR | Zbl
, and , A factorisation of diffusion measure and finite sample path constructions. Methodol. Comput. Appl. Probab. 10 (2008) 85–104. | DOI | MR | Zbl
and , Handbook of Brownian motion – facts and formulae. Probability and its Applications. 2nd ed. Birkhäuser Verlag, Basel (2002). | MR | Zbl
, and , A continuity correction for discrete barrier options, Math. Finance 7 (1997) 325–349. | DOI | MR | Zbl
and , The Theory of Stochastic Processes. John Wiley & Sons Inc, New York (1965). | MR | Zbl
and , The first passage problem for a continuous Markov process, Ann. Math. Stat. 24 (1953) 624–639. | DOI | MR | Zbl
, Nonuniform Random Variate Generation. Springer-Verlag, New York (1986). | DOI | MR | Zbl
and , Asymptotics of two-boundary first-exit-time densities for Gauss–Markov processes. Methodol. Comput. Appl. Probab. 21 (2019) 735–752. | DOI | MR | Zbl
, Weak approximation of killed diffusion using Euler schemes. Stochastic Process. Appl. 87 (2000) 167–197. | DOI | MR | Zbl
and , Stopped diffusion processes: boundary corrections and overshoot. Stochastic Process. Appl. 120 (2010) 130–162. | DOI | MR | Zbl
and , Exact simulation of the first-passage time of diffusions. J. Sci. Comput. 79 (2019) 1477–1504. | DOI | MR | Zbl
and , Diffusion Processes and Their Sample Paths. Second printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 125. Springer-Verlag, Berlin-New York (1974). | MR | Zbl
, Exact simulation of the sample paths of a diffusion with a finite entrance boundary, Preprint (2013). | arXiv
and , Brownian motion and stochastic calculus, 2nd edition. In: Vol. 113 of Graduate Texts in Mathematics. Springer-Verlag, New York (1991). | MR | Zbl
, Exitbm: a library for simulating Brownian motion’s exit times and positions from simple domains. Technical Report INRIA RR-7523 (2011).
and , Simulation of a space-time bounded diffusion. Ann. Appl. Probab. 9 (1999) 732–779. | DOI | MR | Zbl
, and , Joint densities of first hitting times of a diffusion process through two time-dependent boundaries. Adv. Appl. Probab. 46 (2014) 186–202. | DOI | MR | Zbl
and , A Course of Modern Analysis. Cambridge Mathematical Library. Reprint of the fourth (1927) edition. Cambridge University Press, Cambridge (1996). | DOI | MR | Zbl
Cité par Sources :
