## Event Date:

## Event Date Details:

Refreshments served at 3:15 PM

## Event Location:

- South Hall 5607F

Dr. Paolo Cavarani (L'Aquila, Italy)

Title:On Control Invariance methods and their use in Finance

Abstract:Asset transactions in the financial market can be viewed as controls influenc- ing the dynamic evolution of an uncertain system. Uncertainty is traditionally described in terms of a stochastic control process driven by random noise. Re- cent developments in control theory recognized the fact that in physics, biology, engineering, and other fields including finance, probability distributions are es- sentially unknown - or at best largely contingent on ad hoc assumptions - and moved in the direction of obtaining essentially distribution-free results. These developments led to the notion of robustness and robust control-invariance. The unknown random variables affecting the system’s trajectory are assumed to lie in a given set. All prior knowledge of randomness is subsumed under the shape of this set. The ensuing exercise is to determine what properties of the dynamic system remain invariant with respect to that set.

The seminar provides first an introduction to control invariance and to robust control invariance for a simple class of dynamic systems - that of linear systems. The invariance definitions are essentially independent of linearity because in this context the greater complexity of non-linear systems manifests itself more in terms of machinery and computation than in terms of the underlying conceptual framework. Secondly, the invariance notion is generalized to the case of games. Each player in an n-person game is viewed as a source of uncertainty by other players. A multi-invariance notion akin to Nash equilibrium arises when each player succeeds in making a desired property of the system evolution - for example, state membership to a prescribed set - invariant to the strategies of the other players. We call this notion an Invariant Equilibrium (IE). In both cases simple but powerful computational tools based on Lyapunov theory and linear matrix inequalities will be discussed.