An Efficient Algorithm to Simulate a Brownian Motion Over Irregular Domains

Abstract

In this paper, we present an algorithm to simulate a Brownian motion by coupling two numerical schemes: the Euler scheme with the random walk on the hyper-rectangles. This coupling algorithm has the advantage to be able to compute the exit time and the exit position of a Brownian motion from an irregular bounded domain (with corners at the boundary), and being of order one with respect to the time step of the Euler scheme.The efficiency of the algorithm is studied through some numerical examples by comparing the analytical solution with the Monte Carlo solution of some Poisson problems. The Monte Carlo solution of these PDEs requires simulating Brownian motions of different types (natural, reflected or drifted) over an irregular domain.

An integral equation for Root’s barrier and the generation of Brownian increments

“…(the first inequality is sharp but the second is not; see Corollary 4). Such a property is particularly useful in Monte Carlo schemes for computing solutions of PDEs with time-dependent boundaries; similar observations have been made by many different authors before, for example, Milstein and Tretyakov, Deaconu and Hermann, Deaconu, Lejay, and Zein [9,25,34], by using different shapes (e.g., parallelepipeds) than R; however, the above approach via the (SEP µ ) is extremal among these solutions in the sense that it allows one to sample from the arguably simplest distribution for computational purposes U [−1, 1]. It is also clear that Brownian scaling can be used to modify the above algorithm, which is described in detail in Section 4, to sample increments during which the uniform bound is arbitrarily small (i.e., µ = U [−ǫ, ǫ], ǫ > 0).…”

“…While analytic expressions are known, it is not efficient to simulate. This has been pointed out by many authors and the work of Milstein and Tretyakov, Deaconu and Hermann, Deaconu, Lejay, and Zein [9,25,34], proposes the use of exit times of time-space Brownian motion from other shapes than spheres. The approach which is closest to the one presented here is the random "walk over moving spheres" (WoMS) introduced in [9].…”

“…In the short section below we show that the Root solution gives another way to construct such a random walk. It is optimal among all such approaches [9,25,34] in the sense that one samples simply from the uniform distribution. A (theoretical) disadvantage is that the barrier r is not known in explicit form and has to be stored as a table look-up, though the results from the previous sections show that this can be done quite easily.…”

Efficient Monte Carlo Methods for Simulating Diffusion-Reaction Processes in Complex Systems

“…For such structures, it may be more convenient to split the trajectory into "elementary blocks" inside rectangular jump domains rather than spherical ones, as illustrated on Fig. 4 [47,48]. The use of rectangles/parallelepipeds as jump domains allows one to significantly reduce the number of small jumps near the boundary of the medium.…”

Convergence Analysis for Spectral Approximation to a Scalar Transport Equation with a Random Wave Speed

“…In numerical simulation, accounting for uncertainties in input quantities (such as model parameters, initial and boundary conditions, and geometry) becomes an important issue in recent years, especially in risk analysis, safety, and optimal design, see, e.g., [1,7,9,20,23,27]. Many works have been recently devoted to the analysis and the implementation of the Stochastic Galerkin (SG) methods and Stochastic Collocation (SC) techniques for such problems.…”