Weak approximation of stochastic differential equations with sticky boundary conditions

TL;DR

This paper develops and analyzes numerical schemes for weak approximation of sticky SDEs, which model particles with adhesion at boundaries, addressing challenges posed by boundary time spent and enabling solutions to related PDEs.

Contribution

Introduces the first half-order and first-order numerical schemes for sticky SDEs, extending approximation methods to models with boundary adhesion phenomena.

Findings

Schemes achieve theoretical convergence orders supported by numerical experiments.

Algorithms can be applied to solve linear parabolic PDEs with sticky boundary conditions.

Addresses challenges of boundary time spent in scheme design.

Abstract

Sticky diffusion models a Markovian particle experiencing reflection and temporary adhesion phenomena at the boundary. Numerous numerical schemes exist for approximating stopped or reflected stochastic differential equations (SDEs), but this is not the case for sticky SDEs. In this paper, we construct and analyze half-order and first-order numerical schemes for the weak approximation of stochastic differential equations with sticky boundary conditions. We present the algorithms in general setting such that they can be used to solve general linear parabolic partial differential equations with second-order sticky boundary condition via the probabilistic representations of their solutions. Since the sticky diffusion spends non-zero amount of time on boundary, it poses extra challenge in designing the schemes and obtaining their order of convergence. We support the theoretical results with numerical experiments.