Calculus of Variations in Mathematical Finance
Springer 2005 |
ISBN-10: 3540434313 | 120 Зages | PDF | 21 MB
provides an infinite-dimensional differential calculus in the context of
continuous paths stochastic processes.
The calculus includes
formulae of integration by parts and Sobolev spaces of differentiable
functions defined on a probability space. This new book, demonstrating
the relevance of Malliavin calculus for Mathematical Finance, starts
with an exposition from scratch of this theory.
sensitivities) are reinterpreted in terms of Malliavin calculus.
Integration by parts formulae provide stable Monte Carlo schemes for
numerical valuation of digital options. Finite-dimensional projections
of infinite-dimensional Sobolev spaces lead to Monte Carlo computations
of conditional expectations useful for computing American options. Weak
convergence of numerical integration of SDE is interpreted as a
functional belonging to a Sobolev space of negative order. Insider
information is expressed as an infinite-dimensional drift. The last
chapter gives an introduction to the same objects
in the context of jump
processes where incomplete markets appear.