Finite-difference modeling à la Mickens of the distribution of the stopping time in a stochastic differential equation

Abstract

Departing from a general stochastic differential equation with Brownian diffusion, we establish that the distribution of probability of the stopping time is governed by a parabolic partial differential equation. A particular form of the problem under investigation may be associated to a stochastic generalization of the well-known Paris’ law from structural mechanics, in which case, the solution of the boundary-value problem represents the probability distribution of the hitting time. An implicit, convergent and probability-based discretization to approximate the solution of the boundary-value problem is proposed in this work. Using a convenient vector representation of our scheme, we prove that the method preserves the most relevant properties of a probability distribution function, namely, the non-negativity, the boundedness from above by 1, and the monotonicity. In addition, we establish that our method is a convergent technique, and provide some illustrative comparisons against known exact solutions.

Keywords:

• Stochastic differential equations
• nonlinear partial differential equations
• Paris’ equation
• probability distribution of hitting time
• dynamic consistency
• Mickens-type finite-difference method
• analysis of convergence

AMS Subject Classifications:

• 65N06
• 65C20
• 35C05
• 35K20

Acknowledgements

Beforehand, we would like to thank the anonymous reviewers and the associate editor in charge of handling this manuscript for their time and efforts. Their suggestions and criticisms contributed substantially to improve the quality of the present work.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Universidad Autónoma de Aguascalientes through [grant number PIM 14-4].