Not Running in 2015/16
Status for Mathematics students: List D
Commitment: 30 lectures
Assessment: 3 hour exam 100%
Prerequisites: Knowledge of basic stochastic calculus including stochastic differential equations driven by Brownian motion will be assumed. Measure theory and functional analysis are basic tools and familiarity with basic concepts of differentiable manifolds is likely to be needed. Useful preparatory courses include: MA482 Stochastic Analysis, MA460 Differential Geometry.
Content: The natural state space for stochastic differential equations is a smooth manifold. Even if that manifold is a Euclidean space, if the equation has a more interesting structure than that of just additive noise it induces differential geometric structures which help to identify the behaviour of the solutions (the ``volatility'' can often determine a Riemannian metric for example, whose curvature affects the long time behaviour of solutions). On the other hand the solution to the equation can be considered as a map from path space on some , i.e, Wiener space, to the space of the manifold, and this can be analysed by techniques of infinite dimensional calculus, in particular those known as Malliavin Calculus.
The precise content of the course will be decided after consulting those who expect to come to the lectures. If you are intending to come it might help if you could contact me sometime in Term 1. Of course if you have not done so you will be very welcome to come! but you will then have much less influence on the content.
MA 460 looks as if it will be a near perfect course introducing much of the differential geometry which arises in the theory. The first part, at least, is strongly recommended for anyone who wishes to continue in this area either as a researcher or as a practitioner.
Books: The following contain useful material for the course:
Rogers, L. C. G.; Williams, David; Diffusions, Markov processes, and martingales. Vol. 2. It™ calculus. Reprint of the second (1994) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. xiv+480 pp. ISBN: 0-521-77593-0 60J60
Elworthy, David; Geometric aspects of diffusions on manifolds. École d'Eté de Probabilités de Saint-Flour XV-XVII,1985-87, 277-425, Lecture Notes in Math., 1362, Springer, Berlin, 1988.
Bell, Denis R; The Malliavin calculus. Reprint of the 1987 edition. Dover Publications, Inc., Mineola, NY, 2006.
Hsu, Elton P. Stochastic analysis on manifolds. Graduate Studies in Mathematics, 38. American Mathematical Society, Providence, RI, 2002.