Appl Math Optim 54:159–187 (2006)
2006 Springer Science+Business Media, Inc.
Boundary Sensitivities for Diffusion Processes
in Time Dependent Domains
and Nicole El Karoui
Dipartimento di Scienze, Universit`a di Chieti-Pescara,
viale Pindaro 42, 65127 Pescara, Italy
ENSIMAG - INP Grenoble, Laboratoire de Mod´elisation et Calcul - UMR 5523,
IMAG - LMC, BP 53, F 38041 Grenoble Cedex 9, France
Ecole Polytechnique, Centre de Math´ematiques Appliqu´ees,
91128 Palaiseau Cedex, France
Abstract. We study the sensitivity, with respect to a time dependent domain D
of expectations of functionals of a diffusion process stopped at the exit from D
normally reﬂected at the boundary of D
. We establish a differentiability result and
give an explicit expression for the gradient that allows the gradient to be computed
by Monte Carlo methods. Applications to optimal stopping problems and pricing of
American options, to singular stochastic control and others are discussed.
Key Words. Stopped diffusion, Reﬂected diffusion, Time dependent domain, Sen-
sitivity analysis, Monte Carlo methods, Free boundary.
AMS Classiﬁcation. 49Q12, 60J50, 35R35, 60G40.
1.1. Presentation of the Problem and Main Results
In this work we address the problem of the sensitivity of the law of a diffusion process
constrained in a time dependent domain D
, with respect to perturbations of
the domain. Both situations where the process is stopped at the exit from D
the process is normally reﬂected at the boundary are covered. The law of the process is