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__Regularity and convergence rates for SDEs with non-globally Lipschitz coefficients__

S. Cox, M. Hutzenthaler and A. Jentzen

**Abstract**

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.