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A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients. (English) Zbl 1094.60056
The aim of this paper is to study a new type of scheme for the simulation of diffusion processes evolving in one-dimensional discontinuous media. This scheme does not rely on smoothing the coefficients (as classical Euler scheme, Milstein scheme) appearing in the differential operator $$L$$, which generates the diffusion process $$X$$, but uses instead an exact description of the behaviour of their trajectories when they reach the points of discontinuity. The differential operator $$L$$ that generates the diffusion process $$X$$ depends on three piecewise smooth functions, called the characteristic coefficients of $$L$$. These smooth functions may have an infinite number of discontinuities on a countable set of points. The basic idea for the new simulation scheme proceeds as follows: First, the differential operator $$L$$ is replaced by another one whose coefficients are piecewise constant, which provides good approximations of the solutions of the elliptic and parabolic partial differential equations involving $$L$$.
Second, the behaviour of the stochastic process generated by the approximation of $$L$$ at a given time, around discontinuity points of the characteristic coefficients of $$L$$, can be described as a skew Brownian motion. The advantage of using a random walk method is that the time step is incremented with a constant value and not with a random variable, while the algorithm can be implemented locally around the points where discontinuities hold. In the regions where the coefficients are smooth, one may use more efficient algorithms based on classical schemes. Detailed description of the algorithm, its numerical simulation, and local comparison of the trajectories of the diffusion processes with those of a skew Brownian motion are provided. Two examples illustrate the new approach: (a) the doubly skew Brownian motion, and (b) diffusion with a coefficient discontinuous at one point (which is compared with the Euler scheme and with a deterministic scheme).

##### MSC:
 60J60 Diffusion processes 60J65 Brownian motion 65C05 Monte Carlo methods
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