Boundary-preserving numerical schemes for stochastic ordinary and partial differential equations

Doktorsavhandling, 2025

This thesis contributes to the development of boundary-preserving numerical schemes for the strong and weak approximation for stochastic ordinary and partial differential equations (SDEs and SPDEs, respectively). Several of the considered equations model a physical quantity with an inherently restricted range, such as temperature (positive values), stock prices (positive values) or fractions (values in [0,1]), referred to as the invariant domain of the equation. A numerical scheme is said to be boundary-preserving if its numerical approximations are guaranteed to remain within this domain. Boundary preservation is important for the physical interpretability and stability of the numerical approximations. Some established approaches to constructing boundary-preserving schemes are surveyed in the first part of the thesis, and the appended papers explore and develop new methods to guarantee this property.

Paper I combines the Lamperti transform with a Lie--Trotter time splitting to construct a family of boundary-preserving numerical schemes for some scalar SDEs achieving strong convergence of order 1. Paper II constructs boundary-preserving numerical schemes for scalar SDEs by introducing auxiliary stochastic processes to convert the considered SDE into an associated reflected SDE. Paper III constructs a positivity-preserving temporal numerical scheme for some semilinear stochastic heat equations perturbed by temporal white noise. The proposed scheme employs a Lie-–Trotter time splitting method, allowing the deterministic and stochastic parts of the equation to be treated independently. Paper IV combines the ideas from Paper III with a finite difference spatial discretisation to obtain the first positivity-preserving numerical scheme for some semilinear stochastic heat equations perturbed by space-time white noise. Paper V combines the ideas from Paper IV with exact simulation for SDEs to obtain the first boundary-preserving numerical scheme for some semilinear SPDEs perturbed by space-time white noise with bounded invariant domain.