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A Backward Kolmogorov Equation Approach to Compute Means, Moments and Correlations of Path-Dependent Stochastic Dynamical Systems

By Laurent Mertz, NYU Shanghai / University of Nice Sophia Antipolis, Nice

I will present a computational alternative to probabilistic simulations for path-dependent stochastic dynamical systems that are prevalent in engineering mechanics. By way of example, we target (a) stochastic elasto-plasticity (involving transition between elastic and plastic states) and (b) obstacle problems with noise (involving discrete impulses due to collisions with an obstacle). We focus on solving Backward Kolmogorov Equations (BKEs) originating from elasto-plastic and obstacle oscillators. The main challenge in solving BKEs corresponding to these problems is to deal with the non-standard boundary conditions which describe the behavior of the underlying process on the boundary. 
Applications that could make use of this framework abound in many areas of science and technology. This is a joint work with Georg Stadler and Jonathan Wylie.