Feynman-Kac's representation in time-space domains and sensitivity with respect to the domain

Cristina Costantini; Nicole El Karoui; Emmanuel Gobet

doi:10.1016/S1631-073X(03)00364-9

Feynman-Kac's representation in time-space domains and sensitivity with respect to the domain

Cristina Costantini
, Nicole El Karoui
, Emmanuel Gobet

Research output: Contribution to journal › Article › peer-review

Abstract

We consider the solution X = (X_t)_t≥0 of a time-inhomogeneous stochastic differential equation and the exit time τ by (t, X_t)_t≥0 of the time-space domain D. We prove the differentiability of expectations of functionals of X stopped at τ, with respect to the domain D: these results extend those in the literature, known in particular by the analysts for the issues of shape optimization. However from the probabilistic point of view, this is not standard.

Translated title of the contribution	Représentation de Feynman-Kac dans des domaines temps-espace et sensibilité par rapport au domaine
Original language	English
Pages (from-to)	337-342
Number of pages	6
Journal	Comptes Rendus Mathematique
Volume	337
Issue number	5
DOIs	https://doi.org/10.1016/S1631-073X(03)00364-9
Publication status	Published - 1 Sept 2003

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abstract = "We consider the solution X = (Xt)t≥0 of a time-inhomogeneous stochastic differential equation and the exit time τ by (t, Xt)t≥0 of the time-space domain D. We prove the differentiability of expectations of functionals of X stopped at τ, with respect to the domain D: these results extend those in the literature, known in particular by the analysts for the issues of shape optimization. However from the probabilistic point of view, this is not standard.",

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N2 - We consider the solution X = (Xt)t≥0 of a time-inhomogeneous stochastic differential equation and the exit time τ by (t, Xt)t≥0 of the time-space domain D. We prove the differentiability of expectations of functionals of X stopped at τ, with respect to the domain D: these results extend those in the literature, known in particular by the analysts for the issues of shape optimization. However from the probabilistic point of view, this is not standard.

AB - We consider the solution X = (Xt)t≥0 of a time-inhomogeneous stochastic differential equation and the exit time τ by (t, Xt)t≥0 of the time-space domain D. We prove the differentiability of expectations of functionals of X stopped at τ, with respect to the domain D: these results extend those in the literature, known in particular by the analysts for the issues of shape optimization. However from the probabilistic point of view, this is not standard.

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Feynman-Kac's representation in time-space domains and sensitivity with respect to the domain

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