Computation of sensitivities for the invariant measure of a parameter dependent diffusion

Roland Assaraf 1 Benjamin Jourdain 2, 3 Tony Lelièvre 2, 4 Raphaël Roux 5
3 MATHRISK - Mathematical Risk handling
Inria Paris-Rocquencourt, UPEM - Université Paris-Est Marne-la-Vallée, ENPC - École des Ponts ParisTech
4 MATHERIALS - MATHematics for MatERIALS
ENPC - École des Ponts ParisTech, CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique, Inria Paris-Rocquencourt
Abstract : We consider the solution to a stochastic differential equation with a drift function which depends smoothly on some real parameter λ, and admitting a unique invariant measure for any value of λ around λ = 0. Our aim is to compute the derivative with respect to λ of averages with respect to the invariant measure, at λ = 0. We analyze a numerical method which consists in simulating the process at λ = 0 together with its derivative with respect to λ on long time horizon. We give sufficient conditions implying uniform-in-time square integrability of this derivative. This allows in particular to compute efficiently the derivative with respect to λ of the mean of an observable through Monte Carlo simulations.
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Roland Assaraf, Benjamin Jourdain, Tony Lelièvre, Raphaël Roux. Computation of sensitivities for the invariant measure of a parameter dependent diffusion. Stochastics and Partial Differential Equations: Analysis and Computations, A. Debussche; B. Rozovsky, 2017, June 2018, 6 (2), pp.125-183. ⟨10.1007/s40072-017-0105-6⟩. ⟨hal-01192862⟩

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