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Numerical Analysis Of Degenerate Kolmogorov Equations of Constrained Stochastic Hamiltonian Systems

Abstract : In this work, we propose a method to compute numerical approximations of the invariant measures and Rice's formula (frequency of threshold crossings) for a certain type of stochastic Hamiltonian system constrained by an obstacle and subjected to white or colored noise. As an alternative to probabilistic Monte-Carlo simulations, our approach relies on solving a class of degenerate partial differential equations with non-local Dirichlet boundary conditions, as derived in [Mertz, Stadler, Wylie; 2018]. A functional analysis framework is presented; regu-larisation and approximation by the finite element method is applied; numerical experiments on these are performed and show good agreement with probabilistic simulations.
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Laurent Mertz, Olivier Pironneau. Numerical Analysis Of Degenerate Kolmogorov Equations of Constrained Stochastic Hamiltonian Systems. Computers and Mathematics with Applications, Elsevier, 2019, 78 (8), pp.2719-2733. ⟨10.1016/j.camwa.2019.04.013⟩. ⟨hal-02963587⟩

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