Mario Annunziato is researcher in Mathematics, in the field of Numerical Analysis,
at the "Università degli Studi di Salerno" in Italy since the year 2004.

His interests focuse to the numerical solution of Partial Differential
Equations (PDE)
and Integral Equations, related to stochastic processes and stochastic
optimal control, and to modelling and applications of stochastic
processes.

The goals of the research are:

- to find the probability density function(PDF) of a stochastic processes by numerically solving PDEs of parabolic and hyperbolic type, or Volterra Integral equations, by ensuring that the discrete PDF be positive (or monotone) and conservative.
- to develop numerical schemes for optimization problems with PDE constraints, related to the optimal control of stochastic processes.
- to provide equation modelling for random phenomena.

A stochastic processes can be almost completely described by its probability density (transition) function (PDF). The Chapman-Kolmogorov equation is an abstract equation for the PDF. In the case of PDP the CK equation can take the form of system of first order hyperbolic PDEs or Volterra integral equations . These equations sometime are named Liouville Master Equation or generalized Fokker-Planck equations.

The system of hyperbolic PDEs has initial Cauchy conditions and the evolutory PDF solution must be non-negative and total probability conservative in time. Further, depending on the type of point process, the PDEs can have special

PDEs with this problem formulation are very little investigated. The development of stable, positive and mass preserving numerical methods is the aim of this research task.

For stochastic models, in the current scientific literature, the problem is formulated with an average of the cost functional of the stochastic state. The solution of this optimal control problem can be found by solving an Hamilton-Jacobi-Bellman equation.

In this research we propose to use the PDF as representative of the state of the system, define the objective as a functional of the PDF, and use the Fokker-Planck equation as constraint of the optimization problem. This is a new and unexplored framework in the field of optimization.

The solution can be found by formulating the minimization problem as an optimality system of PDEs, in order to find the reduced gradient of the objective and its vanishing point.

The aim of this research is to develop numerical methods to solve the optimality system and special minimization techniques for this non-linear optimization problem.