ASSET ALLOCATION AND ANNUITY-PURCHASE STRATEGIES TO MINIMIZE THE PROBABILITY OF FINANCIAL RUIN

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/75685/1/j.1467-9965.2006.00288.x.pd

Brownian motion and stochastic calculus

This book is designed as a text for graduate courses in stochastic processes. It is written for readers familiar with measure-theoretic probability and discrete-time processes who wish to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed. The power of this calculus is illustrated by results concerning representations of martingales and change of measure on Wiener space, and these in turn permit a presentation of recent advances in financial economics (option pricing and consumption/investment optimization). This book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The text is complemented by a large number of problems and exercises

A decomposition of the Brownian path

The Brownian path {[omega](s); 0 [less-than-or-equals, slant] s [less-than-or-equals, slant] t} is dissected and then reassembled in such a way that (i) the last visit [gamma]t at the origin, as well as the fragment {[omega](s); [gamma]t [less-than-or-equals, slant] s [less-than-or-equals, slant] t}, are left invariant; (ii) on [0, [gamma]t], local time becomes maximum-to-date and occupation time ofR+ becomes location of maximum; and (iii) the resulting process is again Brownian. Characterizations of conditional processes are employed to establish the result. Several consequences of the latter are discussed.60J65 60G17 Brownian motion and bridge local time occupation time conditioning path decomposition