# Athena Picarelli

Position
Temporary Assistant Professor
Academic sector
SECS-S/06 - MATHEMATICAL METHODS OF ECONOMICS, FINANCE AND ACTUARIAL SCIENCES
Research sector (ERC)
35K - Parabolic equations and systems

Office
Polo Santa Marta,  Floor 1,  Room 1.48
Telephone
045 802 8447
E-mail
athenapicarelliunivrit
Personal web page
https://sites.google.com/site/athenapicarelli/

#### Office Hours

Wednesday, Hours 2:30 PM - 4:30 PM,   Polo Santa Marta, Floor 1, room 1.47
Out of the lecture period, please contact me by mail to arrange meetings.

Curriculum

#### Modules

Modules running in the period selected: 3.
Click on the module to see the timetable and course details.

Course Name Total credits Online Teacher credits Modules offered by this teacher
Master’s degree in Banking and Finance Asset Pricing Models (2019/2020)   9
Master’s degree in Economics Mathematical models for business and economics (2019/2020)   6
Master’s degree in Economics Mathematical models for business and economics (2018/2019)   6

Advanced teaching activities
Name Online
Activities PhD Course In Economics and Management (35° ciclo - PhD in Economics and Management)

Skills
Topic Description Research area
MSC 35K61 - Nonlinear initial-boundary value problems for nonlinear parabolic equations The study of parabolic equations is related to evolutive diffusion problems. Hamilton-Jacobi-Bellman equations are fully nonlinear possibly degenerate equations belonging to this class and arise in the study of stochastic optimal control problems. Fixed suitable initial and boundary conditions, I’m interested in the study of existence and uniqueness of solutions, their regularity and numerical approximation. Quantitative Methods for Economics
Parabolic equations and systems
MSC 49L20 - Dynamic programming method An optimal control problem is defined starting from a dynamics, a set of controls acting on this dynamics and a cost (or gain), functional of the control and the associated dynamics. The objective is to minimize (or maximize) this cost (or gain). The value function (function of the initial time and position) is defined as the optimal value, i.e. the minimum (or maximum) value of the cost (or gain), associated to the problem. I study optimal control problems for which the dynamics is given by a stochastic differential equation. By the dynamic programming approach the value function can be characterized as the solution (in the weak viscosity sense) of a partial differential equation called the Hamilton-Jacobi-Bellman equation. Quantitative Methods for Economics
Hamilton-Jacobi theories, including dynamic programming
MSC 65M06 - Finite di erence methods Only in very few cases Hamilton-Jacobi-Bellman equations admit an explicit solution. It becomes then fundamental the numerical approximation of the solution. Numerical methods for partial differential equations are basically divided in: finite elements methods and finite difference methods. The latter are based on a Taylor approximation of derivatives. They are quite simple and intuitive methods for which a complete convergence analysis in the class of solutions of the equation in the viscosity sense is available. Quantitative Methods for Economics
Partial differential equations, initial value and time-dependent initial- boundary value problems
MSC 65M15 - Error bounds Defined a numerical approximation scheme for a partial differential equation and proved its convergence, it is also interesting to provide error estimates. For classical solutions of elliptic and parabolic equation this can be obtained by quite standard techniques. However, in the particular case of viscosity solutions specific analytic regularization techniques have to be applied. Quantitative Methods for Economics
Partial differential equations, initial value and time-dependent initial- boundary value problems
MSC 91G80 - Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) Among the main applications of stochastic optimal control theory one finds mathematical finance. Indeed, many decision problems are formulated in terms of optimization on continuous-time stochastic models. We find typically: hedging problems, portfolio optimization, risk management and optimal stopping. Quantitative Finance
Mathematical finance