**Differential Equations and Optimization in Financial
Markets**

For several decades, the prevailing theory of price dynamics
has been based upon the efficient market hypothesis (EMH) which asserts that
the asset price is determined solely through the information on the asset. A small deviation from the fundamental (or
realistic) value is rapidly corrected through a first order differential
equation that does not allow for oscillations and overshootings of the
fundamental value. The evidence from
both world markets and the newly developing field of experimental markets has
cast doubt on the EMH in some situations.
The large price bubbles and market crashes in recent years have been
among the most dramatic examples.

Research work continues on modeling price dynamics by
deriving systems of nonlinear differential equations that incorporate the
tendency of investors to be influenced by the trend as well as the fundamentals
of an asset. The equations also utilize
natural conservation laws involving total asset and cash supply. The resulting system can then describe both
momentum and liquidity in financial markets.

These equations are being utilized
in conjunction with laboratory asset markets (see summary). This work is in
collaboration with Vladimira Ilieva, David Porter and Vernon Smith.