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abstract:
The main object of this talk is a perturbation formula for stochastic differential equations (SDEs) which expresses the distance between the solution and any It\^o process in terms of the distances of the local characteristics. Under suitable exponential integrability properties, this perturbation formula yields a sufficient condition for local Lipschitz continuity in the strong sense in the initial value, a sufficient condition for explicit numerical approximations of finite-dimensional SDEs and a sufficient condition for spatial discretizations of nonlinear SPDEs. We illustrate these conditions with example SDEs from finance, physics and biology. Finally, we present an example SDE with a globally bounded and smooth drift coefficient and a constant diffusion coefficient whose solution is not locally Hölder continuous in the strong sense in the initial value. |
