The Old General Equilibrium Problem and Some New Mathematics around It

The general equilibrium problem lies at the very foundation of neoclassical economics, and has been extensively studied since 1870s and the work of Leon Walras. The original problem in various formulations inspired a tremendous amount of mathematical theory in areas as diverse as game theory, dynamical systems and topology. Its relationship with stochastic analysis came to its own with the study of stochastic financial-market equilibria in the work of Duffie, Zame, Karatzas, Shreve and Lehoczky and others.

Financial market models can be divided, broadly, into two categories - complete and incomplete. Complete models describe markets which can be used to replicate an insurance contract against any conceivable contingency. Markets in which this, highly unrealistic, assumption is not met are termed incomplete. Thanks to the existence of so-called representative agents, the equilibrium theory of complete markets is significantly easier than its incomplete counterpart, and most problems of existence and characterization have been settled already in 1980s. On the other hand, very little is known, even today, about the incomplete case in continuous time.

Recent developments in stochastic analysis and partial differential equations opened a door for a new approach to this stubborn problem. It led to a positive resolution (jointly with Hao Xing of London School of Economics) in the special case when the economic agents have exponential utility functions and is based on some new results, by the same authors, about systems of quadratic backward stochastic differential equations (BSDEs). It is interesting that these results also cover equations that appear in completely different contexts - e.g., when one tries to construct martingales on Riemannian manifolds or find Nash points of non-zero-sum stochastic games.