Stochastic calculus of variations in mathematical finance

Paul Malliavin, Anton Thalmaier9783540434313, 3-540-43431-3

Malliavin calculus 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.

Greeks (price 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.