Stochastic Dynamics of Financial Markets

The project will focus on the following closely related topics: A. Asset pricing and growth optimal investments under transaction costs. The basis for this line of research is an important class of random dynamical systems introduced by von Neumann and Gale in connection with models of economic growth. Analogies between mathematical frameworks used for the analysis of economic growth and financial dynamics will be exploited to develop a model for a financial market with frictions. Within this model, is sues of pricing and growth will be studied. B. Evolutionary finance. This branch of Mathematical Finance combines behavioral approaches, which attempt to model the psychology of the market, and approaches dealing with equilibrium and dynamics. The mai n emphasis is placed on the question of "survival and extinction" of investment strategies. The following problems will be considered: the characterization of survival strategies, their existence, uniqueness and asymptotic properties; links between surviv al strategies and growth optimal investments; evolutionary asset market games. C. Volatility-induced growth. It has recently been discovered that the volatility of asset prices can serve as an endogenous factor of acceleration (or emergence) of finan cial growth, rather than an impediment to it. One of the goals of the project is to examine this counterintuitive phenomenon from various angles and build a theory explaining it and establishing conditions under which it occurs. Special attention will be devoted to relations between volatility-induced growth and the no-arbitrage hypothesis. The phenomenon of growth acceleration will be examined by invoking such aspects of random dynamical systems as noise-induced stability and stochastic resonance. I n the course of the research along the lines A-C, robust modeling approaches will be pursued, in accordance with the general goals of the project.