SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Regularity and convergence rates for SDEs with non-globally Lipschitz coefficients

S. Cox, M. Hutzenthaler and A. Jentzen

Consider the following SDE in $ \mathbb{R}^d $: $\label{SDE}\tag{SDE} d X_t = A(X_t) dt + B(X_t) dW_t, t > 0; X_0 = x \in \mathbb{R}^d, $ where $W$ is an $m$-dimensional standard Brownian motion, and $ A:\mathbb{R}^d \rightarrow \mathbb{R}^d$ and $B: \mathbb{R}^d \rightarrow \mathbb{R}^{d\times m}$ are continuous mappings. The classical results concerning convergence rates for temporal approximations of this SDE (e.g., convergence rates for the Euler scheme) require the coefficients $A$ and $B$ to be globally Lipschitz continuous. However, many SDEs arising from financial and physical models do not have globally Lipschitz continuous coefficients. In joint work with Martin Hutzenthaler and Arnulf Jentzen, we consider the regularity of the solution to the SDE with respect to the initial value $x$, for certain types of SDEs with non-globally Lipschitz continuous coefficients. We use this to prove convergence rates for approximations.

Organized by         Universidad de Valladolid     IMUVA