Annalisa Cesaroni

Almost Sure Stabilizability of Controlled Degenerate Diffusions

We develop a direct Lyapunov method for the almost sure open-loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes. The infinitesimal decrease condition for a Lyapunov function is a new form of Hamilton--Jacobi--Bellman partial differential inequality of second order. We give local and global versions of the first and second Lyapunov theorems, assuming the existence of a lower semicontinuous Lyapunov function satisfying such an inequality in the viscosity sense. An explicit formula for a stabilizing feedback is provided for affine systems with smooth Lyapunov function. Several examples illustrate the theory.

Convergence by Viscosity Methods in Multiscale Financial Models with Stochastic Volatility

We study singular perturbations of a class of stochastic control problems under assumptions motivated by models of financial markets with stochastic volatilities evolving on a fast time scale. We prove the convergence of the value function to the solution of a limit (effective) Cauchy problem for a parabolic equation of HJB type. We use methods of the theory of viscosity solutions and of the homogenization of fully nonlinear PDEs. We test the result on some financial examples, such as Merton portfolio optimization problem.

A converse Lyapunov theorem for almost sure stabilizability

We prove a converse Lyapunov theorem for almost sure stabilizability and almost sure asymptotic stabilizability of controlled diffusions: given a stochastic system a.s. stochastic open-loop stabilizable at the origin, we construct a lower semicontinuous positive definite function whose level sets form a local basis of viable neighborhoods of the equilibrium. This result provides, with the direct Lyapunov theorems proved in a companion paper, a complete Lyapunov-like characterization of the a.s. stabilizability.

Homogenization of Fronts in Highly Heterogeneous Media

We consider the evolution by mean curvature in a highly heterogeneous medium, modeled by a periodic forcing term, with large

Long-Time Behavior of the Mean Curvature Flow with Periodic Forcing

L^\infty

Optimal Exercise of Swing Contracts in Energy Markets: An Integral Constrained Stochastic Optimal Control Problem ∗

-norm but with zero average. We prove the existence of a homogenization limit, when the dimension of the periodicity cell tends to zero, and show some properties of the effective velocity.

Optimal Control with Random Parameters: A Multiscale Approach

We consider the long-time behavior of the mean curvature flow in heterogeneous media with periodic fibrations, modeled as an additive driving force. Under appropriate assumptions on the forcing term, we show existence of generalized traveling waves with maximal speed of propagation, and we prove the convergence of solutions to the forced mean curvature flow to these generalized waves.

Homogenization and vanishing viscosity in fully nonlinear elliptic equations: rate of convergence estimates

We characterize the value of swing contracts in continuous time as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation with suitable boundary conditions. The case of contracts with penalties is straightforward, and in that case only a terminal condition is needed. Conversely, the case of contracts with strict constraints gives rise to a stochastic control problem with a nonstandard state constraint. We approach this problem by a penalty method: we consider a general constrained problem and approximate the value function with a sequence of value functions of appropriate unconstrained problems with a penalization term in the objective functional. Coming back to the case of swing contracts with strict constraints, we finally characterize the value function as the unique viscosity solution with polynomial growth of the Hamilton-Jacobi-Bellman equation subject to appropriate boundary conditions.

STABILIZATION OF CONTROLLED DIFFUSIONS AND ZUBOV'S METHOD

We model the parameters of a control problem as an ergodic diffusion process evolving at a faster time scale than the state variables. We study the asymptotics as the speed of the parameters gets large. We prove the convergence of the value function to the solution of a limit Cauchy problem for a Hamilton-Jacobi equation whose Hamiltonian is a suitable average of the initial one. We give several examples where the effective Hamiltonian allows to define a limit control problem whose dynamics and payoff are linear or nonlinear averages of the initial data. This is therefore a constant-parameter approximation of the control problem with random entries. Our results hold if the fast random parameters are the only disturbances acting on the system, and then the limit system is deterministic, but also for dynamics affected by a white noise, and then the limit is a controlled diffusion.

Curve shortening flow in heterogeneous media

Abstract This paper is devoted to studying the behavior as ε → 0 of the equations uε + H(x, x/ε, Duε, εγD2uε) = 0 with γ > 0. It is known that, under some periodicity and ellipticity or coercivity assumptions, the solution uε converges to the solution u of an effective equation u + H̅(x, Du) = 0, with an effective Hamiltonian H̅ dependent on the value of γ. The main purpose of this paper is to estimate the rate of convergence of uε to u. Moreover we discuss some examples and model problems.