Hostname: page-component-89b8bd64d-r6c6k Total loading time: 0 Render date: 2026-05-11T15:28:47.190Z Has data issue: false hasContentIssue false
  • English
  • Français

Efficient Estimation of One-Dimensional Diffusion First Passage Time Densities via Monte Carlo Simulation

Part of: Stochastic processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 July 2016

Tomoyuki Ichiba  and
Constantinos Kardaras
Tomoyuki Ichiba*
Affiliation:
University of California Santa Barbara
Constantinos Kardaras*
Affiliation:
Boston University
*
∗ Postal address: Department of Statistics and Applied Probability, University of California Santa Barbara, CA 93106, South Hall 5607A, USA. Email address: ichiba@pstat.ucsb.edu
∗∗ Postal address: Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA 02215, USA. Email address: kardaras@bu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We propose a method for estimating first passage time densities of one-dimensional diffusions via Monte Carlo simulation. Our approach involves a representation of the first passage time density as the expectation of a functional of the three-dimensional Brownian bridge. As the latter process can be simulated exactly, our method leads to almost unbiased estimators. Furthermore, since the density is estimated directly, a convergence of order 1 / √N, where N is the sample size, is achieved, which is in sharp contrast to the slower nonparametric rates achieved by kernel smoothing of cumulative distribution functions.

Keywords

First passage timeMonte Carlo density estimationone-dimensional diffusionthree-dimensional Brownian bridgerate function

MSC classification

Secondary: 65C05: Monte Carlo methods 60G44: Martingales with continuous parameter

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011 , pp. 699 - 712
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Alili, L. and Patie, P. (2010). Boundary-crossing identities for diffusions having the time-inversion property. J. Theoret. Prob. 23, 65–84.Google Scholar
[2] Baldi, P., Caramellino, L. and Iovino, M. G. (1999). Pricing general barrier options: a numerical approach using sharp large deviations. Math. Finance 9, 293–322.Google Scholar
[3] Berman, S. M. (1985). An asymptotic bound for the tail of the distribution of the maximum of a Gaussian process. Ann. Inst. H. Poincaré Prob. Statist. 21, 47–57.Google Scholar
[4] Durbin, J. (1985). The first-passage density of a continuous Gaussian process to a general boundary. J. Appl. Prob. 22, 99–122.Google Scholar
[5] Elliott, J. (1955). Eigenfunction expansions associated with singular differential operators. Trans. Amer. Math. Soc. 78, 406–425.Google Scholar
[6] Elworthy, K. D., Li, X.-M. and Yor, M. (1999). The importance of strictly local martingales; applications to radial Ornstein–Uhlenbeck processes. Prob. Theory Relat. Fields 115, 325–355.Google Scholar
[7] Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering (Appl. Math. 53). Springer, New York.Google Scholar
[8] Gobet, E. (2000). Weak approximation of killed diffusion using Euler schemes. Stoch. Process. Appl. 87, 167–197.Google Scholar
[9] Göing-Jaeschke, A. and Yor, M. (2003). A clarification note about hitting times densities for Ornstein–Uhlenbeck processes. Finance Stoch. 7, 413–415.Google Scholar
[10] Hinton, D. and Schaefer, P. W. (eds) (1997). Spectral Theory and Computational Methods of Sturm–Liouville Problems (Lecture Notes Pure Appl. Math. 191). Marcel Dekker, New York.Google Scholar
[11] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus (Graduate Texts Math. 113), 2nd edn. Springer, New York.Google Scholar
[12] Kent, J. T. (1980). Eigenvalue expansions for diffusion hitting times. Z. Wahrscheinlichkeitsth. 52, 309–319.Google Scholar
[13] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations (Appl. Math. 23). Springer, Berlin.Google Scholar
[14] Lehmann, A. (2002). Smoothness of first passage time distributions and a new integral equation for the first passage time density of continuous Markov processes. Adv. Appl. Prob. 34, 869–887.Google Scholar
[15] Milstein, G. N. and Tretyakov, M. V. (2004). Evaluation of conditional Wiener integrals by numerical integration of stochastic diferential equations. J. Comput. Phys. 197, 275–298.Google Scholar
[16] Milstein, G. N. and Tretyakov, M. V. (2004). Stochastic Numerics for Mathematical Physics. Springer, Berlin.Google Scholar
[17] Pauwels, E. J. (1987). Smooth first-passage densities for one-dimensional diffusions. J. Appl. Prob. 24, 370–377.Google Scholar
[18] Peskir, G. (2002). On integral equations arising in the first-passage problem for Brownian motion. J. Integral Equations Appl. 14, 397–423.Google Scholar
[19] Pinsky, R. G. (1995). Positive Harmonic Functions and Diffusion. (Camb. Stud. Adv. Math. 45). Cambridge University Press.Google Scholar
[20] Pitman, J. and Yor, M. (1999). The law of the maximum of a Bessel bridge. Electron. J. Prob. 4, 35pp.Google Scholar
[21] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion. (Fundamental Principles Math. Sci. 293), 3rd edn. Springer, Berlin.Google Scholar
[22] Rogers, L. C. G. (1985). Smooth transition densities for one-dimensional diffusions. Bull. London Math. Soc. 17, 157–161.Google Scholar
[23] Zettl, A. (2005). Sturm-Liouville Theory (Math. Surveys Monogr. 121). American Mathematical Society, Providence, RI.Google Scholar