A Strongly Monotonic Polygonal Euler Scheme

TL;DR

This paper introduces a novel explicit truncation method for monotonic functions in Hilbert spaces, enabling the construction of a polygonal Euler scheme that preserves monotonicity in SDE simulations, without complex optimization.

Contribution

It presents the first explicit, infinite-dimensional truncation method for monotone functions, facilitating monotonicity-preserving schemes for stochastic differential equations.

Findings

The method is well-defined with minimal assumptions.

It preserves the monotonicity of the drift coefficient.

It avoids solving optimization problems like Moreau-Yosida regularisation.

Abstract

In recent years tamed schemes have become an important technique for simulating SDEs and SPDEs whose continuous coefficients display superlinear growth. The taming method, which involves curbing the growth of the coefficients as a function of stepsize, has so far however not been adapted to preserve the monotonicity of the coefficients. This has arisen as an issue particularly in \cite{articletam}, where the lack of a strongly monotonic tamed scheme forces strong conditions on the setting. In the present work we give a novel and explicit method for truncating monotonic functions in separable Hilbert spaces, and show how this can be used to define a polygonal (tamed) Euler scheme on finite dimensional space, preserving the monotonicity of the drift coefficient. This new method of truncation is well-defined with almost no assumptions and, unlike the well-known Moreau-Yosida regularisation, does not require an optimisation problem to be solved at each evaluation. Our construction is the first infinite dimensional method for truncating monotone functions that we are aware of, as well as the first explicit method in any number of dimensions.