TALK BY Mykhaylo Shkolnikov, April 14, 2011
Large systems of interacting diffusion processes
We will consider two systems of interacting diffusion processes,
which go by the names rank-based and volatility-stabilized models in the
mathematical finance literature. We will show that, if one lets the number of
diffusion processes tend to infinity, the limiting dynamics of the system is
described by the porous medium equation with convection in the rank-based case
and by a degenerate linear parabolic equation in the volatility-stabilized
case. In the first case we also provide the corresponding large deviations
principle. The results can be applied in stochastic portfolio theory and for
the numerical solution of partial differential equations. A part of the talk is
joint work with Amir Dembo and Ofer Zeitouni.