The following gives a brief breakdown of the main areas of activity for the Symposium.
theory of optimal stopping
stopping and free boundary problems
(1) The principles of smooth and continuous fit. Recent advances in clarification of the two most useful analytic facts are to be exposed. Illuminating discussions on this topic are anticipated.
(2) The maximality principle. The intriguing problem is to examine under what conditions this principle remains valid for Markov processes with jumps. Some of the recent work has focused on clarifying this issue in some special cases of Lévy processes.
(3) Fluctuation theory of Lévy processes and applications to optimal stopping. Many solutions to optimal stopping problems boil down to a first passage problem of an underlying Markov process. The recent advances in the theory of Lévy processes, and in particular fluctuation theory, has also provided a cache of knowledge offering new possibilities for solving optimal stopping problems driven by Lévy processes.
(4) Stochastic analysis and sharp inequalities. The most challenging problem remains to determine the best constant in the classic Burkholder-Davis-Gundy inequality. Some recent progress in this direction is to be disclosed.
(1) A number of recent contributions to the pricing of American type options in Black-Scholes and other semi-martingale markets (in particular Lévy driven markets) will be exposed. Examples of such options include the classical American put (for which new results are still being established), Russian options, Asian options, Integral options, Bermudan options and so on. Attention will be drawn to accurate characterizations of optimal exercise boundaries, early exercise decompositions, and associated free boundary problems.
(2) Credit risk has recently shown that the issue of establishing endogenous default levels for the equity of a firm boils down to understanding a particular class of optimal stopping problems. This most recent direction and its intimate relationship with smooth and continuous fit principles will be considered.
(3) Game type options are a natural extension of American type options. These are options in which both writer and seller have the right to exercise, each resulting in a pay out for the holder. The writer's objective to minimize the payout and the holder to maximize. A number of recent examples of such options have been considered both with finite and infinite time horizons. Results in this direction as well as the many challenges that lay ahead will be presented here.
(1) Sequential testing. The old problem of testing three simple hypotheses still remains unsolved. It is hoped that the recent advances in characterizing the optimal boundaries as unique solutions of nonlinear integral equations may be helpful in tackling the problem.
(2) Quickest detection. Recent applications involve a quickest detection of arbitrage. A significant progress in understanding the problem is achieved for Markov processes with jumps and the exchange of facts and ideas in this direction will be most stimulating and useful.
(3) Optimal prediction. This is a
new problem formulation in continuous time where the underlying gain process
is not adapted. Recently developed techniques to solve problems of this
type are to be exposed. New optimal prediction problems are to be formulated.
These problems appear useful in diverse applications, most notably in financial
mathematics and financial engineering.