7,008 research outputs found
Feynman-Kac representation of fully nonlinear PDEs and applications
The classical Feynman-Kac formula states the connection between linear
parabolic partial differential equations (PDEs), like the heat equation, and
expectation of stochastic processes driven by Brownian motion. It gives then a
method for solving linear PDEs by Monte Carlo simulations of random processes.
The extension to (fully)nonlinear PDEs led in the recent years to important
developments in stochastic analysis and the emergence of the theory of backward
stochastic differential equations (BSDEs), which can be viewed as nonlinear
Feynman-Kac formulas. We review in this paper the main ideas and results in
this area, and present implications of these probabilistic representations for
the numerical resolution of nonlinear PDEs, together with some applications to
stochastic control problems and model uncertainty in finance
G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Ito's type
We introduce a notion of nonlinear expectation --G--expectation-- generated
by a nonlinear heat equation with infinitesimal generator G. We first discuss
the notion of G-standard normal distribution. With this nonlinear distribution
we can introduce our G-expectation under which the canonical process is a
G--Brownian motion. We then establish the related stochastic calculus,
especially stochastic integrals of Ito's type with respect to our G--Brownian
motion and derive the related Ito's formula. We have also give the existence
and uniqueness of stochastic differential equation under our G-expectation. As
compared with our previous framework of g-expectations, the theory of
G-expectation is intrinsic in the sense that it is not based on a given
(linear) probability space.Comment: Submited to Proceedings Abel Symposium 2005, Dedicated to Professor
Kiyosi Ito for His 90th Birthda
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