Abstracts

A new class of explicit Euler schemes, which approximate stochastic differential equations (SDEs) with superlinearly growing drift and diffusion coefficients, will be presented in this talk. It will be shown that, under very mild conditions, these explicit schemes converge in probability and in $\mathcal L^p$ to the solution of the corresponding SDEs. Key ideas from [1] and [4] which are used for the derivation of the aforementioned results will be highlighted. Moreover, rate of convergence estimates will be provided for strong and almost sure convergence. In particular, it will be shown that the strong order 1/2 is recovered in the case of uniform $\mathcal L^p$-convergence. Finally, a comparison will be made with the most recent developments in the field, namely tamed Euler (see [2] and [3]) and balanced (see [5]) methods. One of the examples will be the popular $3/2$ stochastic volatility model in Finance. Jump processes are also considered.

[1] I. Gyöngy, S. Sabanis, A note on Euler approximations for stochastic differential equations with delay, Appl. Math. Optim., 68 (2013), no. 3, pp. 391--412.
[2] Hutzenthaler, M. and Jentzen, A.: Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, \emph{ArXiv} \url{arXiv:1203.5809 [math.PR]}.
[3] M. Hutzenthaler, A. Jentzen, P.E. Kloeden, Strong convergence of an explicit numerical method for SDEs with non-globally Lipschitz continuous coefficients. Ann. Appl. Probab. 22 (2012) 1611--1641.
[4] S. Sabanis, A note on tamed Euler approximations, Electron. Commun. Probab. 18 (2013), no. 47, 1–-10.
[5] M.V. Tretyakov and Z. Zhang, A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications, SIAM J. Numer. Anal. 51(2013), no. 6, 3135–-3162.