Boundary-preserving Lamperti--It\^o--Taylor approximations for some stochastic differential equations

In this work, we propose high-order boundary-preserving numerical schemes for the strong approximation for some scalar stochastic differential equations with invariant domains being open and bounded intervals.

The proposed methods involve using the Lamperti transform to map the SDE to another SDE with additive noise with a trivial invariant domain.

Then, by imposing regularity assumptions on the original coefficient functions, we can guarantee that the drift coefficient function of the transformed SDE is regular, and known high-order schemes can be used to achieve the desired convergence order.

We confirm the theoretical results with numerical experiments.