WHITE NOISE THEORY WITH APPLICATIONS

In stochastic calculus "white noise" is usually considered an ill-defined object, which in some sense is supposed to be the (non-existing!) time-derivative of Brownian motion. Accordingly, to avoid this object mathematicians often rewrite stochastic differential equations (which involve white noise) as stochastic integral equations, where only Ito integrals (with respect to the familiar and commonly accepted Brownian motion) are used, and then apply Ito calculus.

This works fine when the quantities involved are adapted and the equations are ordinary stochastic differential equations. However, for many stochastic partial differential equations (SPDEs) it is necessary to put everything into a framework where white noise is a well-defined object and where operations with white noise (addition, multiplication...) can be performed more freely (and with a clear conscience).

The white noise theory achieves all this in an elegant way. Moreover, not only does this theory enable us to solve a number of SPDEs - it also gives us a new and elegant approach to Ito/Skorohod integration, Malliavin derivative, Clark-Haussmann-Ocone formula and other results which have many applications, e.g. to economics. In addition, it points out how we can obtain similar results for other types of noise (e.g. fractional and Poissonian).

In the course we will (if time permits)

1) first give a short introduction to the fundamentals of white noise

theory and then proceed to some of the applications:

2) SPDEs, possibly including the Zakai equation in nonlinear filtering

theory and the Musiela equation in mathematical finance

3) Malliavin calculus, the generalized Clark-Haussmann-Ocone formula

and applications to hedging in finance

4) Other types of noise, e.g. the noise of a Poisson process.

The course will be based on the book:

H.Holden et al: Stochastic Partial Differential Equations. Birkhäuser 1996.

Time: Fridays 8.15 - 10.00 (NB!)

Place: Seminar room B62 in the N. H. Abel Building, Blindern.

First time: 26 January 2001.

Fred Espen Benth Bernt Øksendal