Paolo Dai Pra

Connections between stochastic control and dynamic games

We consider duality relations between risk-sensitive stochastic control problems and dynamic games. They are derived from two basic duality results, the first involving free energy and relative entropy and resulting from a Legendre-type transformation, the second involving power functions. Our approach allows us to treat, in essentially the same way, continuous- and discrete-time problems, with complete and partial state observation, and leads to a very natural formal justification of the structure of the cost functional of the dual. It also allows us to obtain the solution of a stochastic game problem by solving a risk-sensitive control problem.

STATIONARY MEASURES AND PHASE TRANSITION FOR A CLASS OF PROBABILISTIC CELLULAR AUTOMATA

We discuss various properties of Probabilistic Cellular Automata, such as the structure of the set of stationary measures and multiplicity of stationary measures (or phase transition) for reversible models.

Pathwise Optimality in Stochastic Control

We introduce a notion of pathwise optimality for stochastic control problems over an infinite time horizon, and give sufficient conditions for the existence of pathwise optimal controls. We analyze both diffusion processes and processes with discrete state space.

A Simple Mean Field Model for Social Interactions: Dynamics, Fluctuations, Criticality

We study the dynamics of a spin-flip model with a mean field interaction. The system is non reversible, spacially inhomogeneous, and it is designed to model social interactions. We obtain the limiting behavior of the empirical averages in the limit of infinitely many interacting individuals, and show that phase transition occurs. Then, after having obtained the dynamics of normal fluctuations around this limit, we analyze long time fluctuations for critical values of the parameters. We show that random inhomogeneities produce critical fluctuations at a shorter time scale compared to the homogeneous system.

Scaling and multiscaling in financial series: a simple model

We propose a simple stochastic volatility model which is analytically tractable, very easy to simulate, and which captures some relevant stylized facts of financial assets, including scaling properties. In particular, the model displays a crossover in the log-return distribution from power-law tails (small time) to a Gaussian behavior (large time), slow decay in the volatility autocorrelation, and multiscaling of moments. Despite its few parameters, the model is able to fit several key features of the time series of financial indexes, such as the Dow Jones Industrial Average, with remarkable accuracy.

Logarithmic Sobolev Inequality for Zero-Range Dynamics

We prove that the logarithmic Sobolev constant for zero-range processes in a box of diameter

Entropy decay for interacting systems via the Bochner-Bakry-Émery approach

L

First occurrence time of a large density fluctuation for a system of independent random walks

grows as

On dynamic programming for sequential decision problems under a general form of uncertainty

L^2

Sharp estimates for the occurrence time of rare events for symmetric simple exclusion

.