Teaching code

4S001444

Teacher

Coordinatore

Viorel Barbu

Credits

6

Scientific Disciplinary Sector (SSD)

MAT/06 - PROBABILITY AND STATISTICS

Language

English

Period

II sem. dal Mar 1, 2017 al Jun 9, 2017.

Moodle Seminars

Learning outcomes

This course is devoted to the presentation of basic results of theory of stochastic differential equations and their models in applied sciences.
The necessary prerequisites of Probability and Real Analysis for this course are briefly reviewed in the first courses.

Program

1. Review of stochastic analysis
1.1 Stochastic processes
1.2 Brownian motion
1.3 The It^o integral
1.4 The It^o formula
2. Stochastic differential equations (SDEs)
2.1 Existence theorems for strong solutions
2.2 Martingale (weak) solutions
2.3 Linear SDEs
2.4 Backward stochasstic differential equations
2.5 Explicit solutions to SDE
3. SDEs in Physics and Economics
3.1 Langevin 's equation
3.2 The Ornstein-Uhlenbeck process
3.3 The geometric Brownian motion equations
3.4 The Black & Scholes model for the stock price model
3.5 Population growth model
4. The It^o diffusion
4.1 The Kolmogorov equation for It^o's diffusion
4.2 Examples to stock-price equation, Ornstein-Uhlenbeck equation
4.3 The Feynmann-Kac formula
4.4 The Fokker-Planck equation
5. Stochastic optimal control problems
5.1 Open loop stochastic opotimal control problems; necessary conditions
of optimality
5.2 Stochastic optimal feedback controllers
5.3 The dynamic programming equation
5.4 The linear stochastic regulator problem
6. Stochastic optimal control in Economics and Finance
6.1 The stochastic production inventory problem
6.2 Optimal portfolio selection problem
6.3 The Merton model of optimal portfolio
6.4 The stochastic advertizing problem
7. Optimal stopping
7.1 It^o's formula with stopping times
7.2 The Dirichlet problem { stochastic approach
7.3 The optimal stopping problem
7.4 Example to American option price
8. Stochastic partial differential equations
8.1 The reaction-diffusion equation
8.2 The stochastic heat equation in Rn
8.3 The stochastic wave equation
8.4 Example to population dynamics and neurophysiology
Bibliography (selective)
[1] L.C. Evans, An Introduction to Stochastic Differential Equations,
U.C. Berkeley, 2000.
[2] B. Oksendall, Stochastic Differential Equations, Springer, 2000.
[3] W. Fleming, R. Rishel, Deterministic and Stochastic Optimal Control,
Springer-Verlag, 1975.

Examination Methods