On Markov chain approximations for computing boundary crossing probabilities of diffusion processes

Abstract

We propose a discrete time discrete space Markov chain approximation with a Brownian bridge correction for computing curvilinear boundary crossing probabilities of a general diffusion process on a finite time interval. For broad classes of curvilinear boundaries and diffusion processes, we prove the convergence of the constructed approximations in the form of products of the respective substochastic matrices to the boundary crossing probabilities for the process as the time grid used to construct the Markov chains is getting finer. Numerical results indicate that the convergence rate for the proposed approximation with the Brownian bridge correction is O(n −2 ) in the case of C 2 -boundaries and a uniform time grid with n steps.

On extension of the Markov chain approximation method for computing Feynman--Kac type expectations

“…Section 3 in [5]). Methods for solving the latter problem are numerous, and we refer the interested reader to [13] for a literature review. To extend the aforementioned problem to the case of stochastic interest rates, one needs to introduce a numeraire process.…”

“…The purpose of this paper is to extend the Markov chain approximation method proposed in [13] to compute the above expectation in the case when the numeraire process takes the form N(T ) = exp{ T 0 V (X(t)) dt} for some continuous function V , i.e. to evaluate…”

“…The method suggested in [13] was a development of the approach from [9] for computing boundary non-crossing probabilities of the Brownian motion process. That approach suggested to replace the continuous dynamics of the underlying Brownian motion process with those of a denumerable discrete time Markov chain, by discretising both time and space using uniform grids.…”

“…The desired approximation for the boundary non-crossing probability was then computed in [9] by multiplying finite-dimensional substochastic matrices obtained by retaining the entries corresponding to the space nodes located between the given boundaries at the respective times. Our modification of this method developed in [13] dramatically improved the convergence rate of the procedure by shifting and adjusting the uniform space grids so that they would have nodes lying exactly on the boundaries, and by implementing one-step Brownian bridge corrections taking into account the possibility that the process could cross a boundary at a time between the points on the discrete time grid. At the same time, the method was extended in [13] to a much broader class of diffusions.…”

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An efficient discrete time and space Markov chain approximation employing a Brownian bridge correction for computing curvilinear boundary crossing probabilities for general diffusion processes was recently proposed in Liang and Borovkov (2021). One of the advantages of that method over alternative approaches is that it can be readily extended to computing expectations of path-dependent functionals over the event of the process trajectory staying between two curvilinear boundaries.

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