A strongly monotonic polygonal Euler scheme

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 involves curbing the growth of the coefficients as a function of stepsize, but so far has not been adapted to preserve the monotonicity of the coefficients. This has arisen as an issue in [4], where the lack of a strongly monotonic tamed scheme forces strong conditions on the setting. In this article we give a novel and explicit method for truncating monotonic functions in separable real 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, and converging to the true solution at the same rate as the classical Euler scheme for Lipschitz coefficients. Our construction is the first explicit method for truncating monotone functions we are aware of, and the first in infinite dimensions.