388 research outputs found
Functional It\^{o} calculus and stochastic integral representation of martingales
We develop a nonanticipative calculus for functionals of a continuous
semimartingale, using an extension of the Ito formula to path-dependent
functionals which possess certain directional derivatives. The construction is
based on a pathwise derivative, introduced by Dupire, for functionals on the
space of right-continuous functions with left limits. We show that this
functional derivative admits a suitable extension to the space of
square-integrable martingales. This extension defines a weak derivative which
is shown to be the inverse of the Ito integral and which may be viewed as a
nonanticipative "lifting" of the Malliavin derivative. These results lead to a
constructive martingale representation formula for Ito processes. By contrast
with the Clark-Haussmann-Ocone formula, this representation only involves
nonanticipative quantities which may be computed pathwise.Comment: Published in at http://dx.doi.org/10.1214/11-AOP721 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The vanishing of L2 harmonic one-forms on based path spaces
We prove the triviality of the first L2 cohomology class of based path spaces
of Riemannian manifolds furnished with Brownian motion measure, and the
consequent vanishing of L2 harmonic one-forms. We give explicit formulae for
closed and co-closed one-forms expressed as differentials of functions and
co-differentials of L2 two-forms, respectively; these are considered as
extended Clark-Ocone formulae. A feature of the proof is the use of the
temporal structure of path spaces to relate a rough exterior derivative
operator on one-forms to the exterior differentiation operator used to
construct the de Rham complex and the self-adjoint Laplacian on L2 one-forms.
This Laplacian is shown to have a spectral gap
Stochastic analysis for obtuse random walks
We present a construction of the basic operators of stochastic analysis
(gradient and divergence) for a class of discrete-time normal martingales
called obtuse random walks. The approach is based on the chaos representation
property and discrete multiple stochastic integrals. We show that these
operators satisfy similar identities as in the case of the Bernoulli randoms
walks. We prove a Clark-Ocone-type predictable representation formula, obtain
two covariance identities and derive a deviation inequality. We close the
exposition by an application to option hedging in discrete time.Comment: 26 page
Poisson stochastic integration in Banach spaces
We prove new upper and lower bounds for Banach space-valued stochastic
integrals with respect to a compensated Poisson random measure. Our estimates
apply to Banach spaces with non-trivial martingale (co)type and extend various
results in the literature. We also develop a Malliavin framework to interpret
Poisson stochastic integrals as vector-valued Skorohod integrals, and prove a
Clark-Ocone representation formula.Comment: 26 page
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