Lower Error Bounds for Strong Approximation of Scalar SDEs with non-Lipschitzian Coefficients
Provides theoretical lower bounds for a broad class of SDEs, establishing fundamental limits for numerical methods based on Brownian motion evaluations.
The paper proves lower error bounds for strong approximation of scalar SDEs with non-Lipschitzian coefficients, showing that known bounds for globally regular coefficients extend to locally regular ones. The bounds are sharp for many cases including Cox-Ingersoll-Ross processes and equations with superlinear or discontinuous coefficients.
We study pathwise approximation of scalar stochastic differential equations at a single time point or globally in time by means of methods that are based on finitely many observations of the driving Brownian motion. We prove lower error bounds in terms of the average number of evaluations of the driving Brownian motion that hold for every such method under rather mild assumptions on the coefficients of the equation. The underlying simple idea of our analysis is as follows: the lower error bounds known for equations with coefficients that have sufficient regularity globally in space should still apply in the case of coefficients that have this regularity in space only locally, in a small neighborhood of the initial value. Our results apply to a huge variety of equations with coefficients that are not globally Lipschitz continuous in space including Cox-Ingersoll-Ross processes, equations with superlinearly growing coefficients, and equations with discontinuous coefficients. In many of these cases the resulting lower error bounds even turn out to be sharp.