Section: New Results

Models and simulations for skew diffusion

Simulating Diffusion Processes in Discontinuous Media: Benchmark Tests

Participant : Géraldine Pichot.

Grants: H2MN04 9.1.1

Software: SBM 6.6

Publications: submitted.

Abstract: We present several benchmark tests for Monte Carlo methods for simulating diffusion in one-dimensional discontinuous media, such as the ones arising the geophysics and many other domains. These benchmarks tests are developed according to their physical, statistical, analytic and numerical relevance. We then perform a systematic study on four numerical methods.

One-dimensional skew diffusions: explicit expressions of densities and resolvent kernel

Participants : Lionel Lenôtre, Géraldine Pichot.

Grants: H2MN04 9.1.1

Publications: [31]

Abstract: The study of skew diffusion is of primary concern for their implication in the modeling and simulation of diffusion phenomenons in media with interfaces. First, we provide results on one-dimensional processes with discontinuous coefficients and their connections with the Feller theory of generators as well as the one of stochastic differential equations involving local time. Second, in view of developing new simulation techniques, we give a method to compute the density and the resolvent kernel of skew diffusions. Explicit closed-form are given for some particular cases.

Algorithms for the simulation of Feller processes

Participant : Lionel Lenôtre.

Grants and projects: H2MNO4 9.1.1 .

Publications: [34] .

Abstract: Two new numerical schemes are created for Skew Diffusions processes. Both algorithms rely on a more generic numerical scheme that can be used for any kind of Feller processes. The proof of convergence for this generic numerical scheme is performed.

Theoretical results on multidimensional Skew Diffusions

Participant : Lionel Lenôtre.

Grants and projects: H2MNO4 9.1.1 .

Publications: [33] .

Abstract: Some significant results on the distribution of the marginal processes of multidimensional Skew Diffusions are found together with new formula. In addition, totally analytical proofs of some results and algorithms given by A. Lejay are given.