New Monte Carlo schemes for simulating diffusions in discontinuous media

Anomaly Detection in Random Heterogeneous Media

“…In the last decade, the numerical simulation of diffusion processes with discontinuous coefficients has been the subject of active research, cf., e.g., [25,45,46,47,106,108,110,116,117,120,122]. While the one-dimensional situation is well understood, the question whether there exist exact simulation schemes for the multi-dimensional case is still unsettled.…”

“…The following approximation of (3.18) to simulate Y t+δt from Y t was proposed in [108]: Since the processes X and W (2) are independent, the law of Y t conditioned on Y 0 = y and X follows a Gaussian distribution with mean y and variance…”

“…In this paragraph, we consider the simulation in the special case of digitized media. More precisely, we consider the mixing scheme by Lejay and Maire [108] which is based on exact simulation of excursions of the one-dimensional skew Brownian motion.…”

“…Note that in contrast to solutions to stochastic differential equations with smooth coefficients, which can be simulated very efficiently, see, e.g., Kloeden and Platen [89], the diffusion processes we are interested in here evolve in discontinuous (digitized) media. In the last decade, the numerical simulation of such processes has been the subject of active research, cf., e.g., [25,45,46,47,106,108,110,120,122]. While the one-dimensional situation is now well understood, the question whether there exist exact simulation schemes for the multi-dimensional case is still unsettled.…”

“…In this work, we study the special case of digitized media. The proposed simulation method uses a scheme introduced by Lejay and Maire [108] based on exact simulation of excursions of the one-dimensional skew Brownian motion. We provide a convergence analysis for the corresponding continuum micro-scale Monte Carlo method by generalizing a classical argument due to Kipnis and Varadhan [86].…”