A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients

Abstract

The aim of this article is to provide a scheme for simulating diffusion processes evolving in one-dimensional discontinuous media. This scheme does not rely on smoothing the coefficients that appear in the infinitesimal generator of the diffusion processes, but uses instead an exact description of the behavior of their trajectories when they reach the points of discontinuity. This description is supplied with the local comparison of the trajectories of the diffusion processes with those of a skew Brownian motion.

Simulating diffusions with piecewise constant coefficients using a kinetic approximation

“…Eq. (2) is obtained by a simple generalization of the computations of [27] (See also [30] for example) which are themselves relying on arguments of J.-F. Le Gall to prove existence and uniqueness of solutions of SDE with local time [26].…”

“…See however, a recent article from N. Limic [31]. Several algorithms have then been proposed to simulate onedimensional diffusion process with discontinuous coefficients [30,18,17,16,35,34,13].…”

A Donsker theorem to simulate one-dimensional processes with measurable coefficients

“…and where L x t (X) is the symmetric local time of X in x (see [LM06]). We see that X behaves like the Brownian motion W except in −1/2 and 1/2.…”

“…In [LM06] (see also [Mar04]) A. Lejay and M. Martinez propose a different scheme. First they perform a piecewise constant approximation of the coefficients.…”

“…This corresponds to the average time spent by Y on each cell of Φ(g). The obtained algorithm is easy to implement, fast, and allows to treat the same cases as in [LM06] except that the initial condition has to be taken in W 1,∞ ∩ H 1 ∩ C 0 . Notice that the convergence rate is of order O(h 1/2 ), which is not surprising comparing to Donsker's theorem.…”