## Program in Probability and Its Applications

Winter Kolmogorov Lecture Series

Tuesday, January 26, 1999
Hans Follmer

*(Humboldt Universitat - Berlin) * *Probabilistic Problems
Arising From Finance*
This lecture will be held at the ROM. Please see below. We review some recent
developments in Probability which are motivated by problems of hedging derivatives
in incomplete financial markets. This will include new variants of decomposition
theorems for semimartingales, the construction of efficient hedges which minimize
the shortfall risk under some cost constraint, and some results on Brownian
motion related to the heterogeneity of information among financial agents.

*ROM
Theatre is at 100 Queen's Park, Toronto. This lecture will be held during the
Probability in Finance Workshop.

Tuesday, February 9, 1999
Burgess Davis

*(Purdue University) * *Perturbed and Reinforced Random
Walks*
Suppose we have two coins, one fair and the other with probability *p*
of heads and 1-*p* of tails. Let X0, X1, X2, .... be the random walk constructed
as follows. Put X0 = 0, and for *n* > 0, toss one of the coins to decide
whether X*n* is one greater (heads) or one less than Xn - 1. For the first
jump (*n*= 1),make this toss with the fair coin. For the other jumps, toss
the fair coin if min{X*k* : *k* < *n*} < X*n*-1 <
max{X*k* : *k* < *n*}, and otherwise toss the *p* coin.
In other words, only use the *p* coin if the walker is at the edge of the
interval of visited sites. If *p* is not equal to 1/2, the jump probabilities
depend on the history of the walk, so the walk is not Markov, and many of the
tools used to study fair random walk (the *p* = 1/2 case) are not applicable.
I will discuss what is known about these and some related random walks.

Tuesday, March 23, 1999
Krzysztof Burdzy

*(University of Washington) * *Hot Bodies*
Where is the hottest spot in a hot body? Is it on the surface or is it hidden
below it? I will report on the more than skin deep research on these questions,
performed jointly with R. Banuelos, W. Werner and R. Bass. The analytic results
described were proved using a probabilistic coupling technique, previously developed
in collaboration with W. Kendall for a statistically motivated project.