46123 - Stochastic Differential Equations

Learning outcomes

At the end of the course, the student will know the basics of the theory of stochastic differential equations and their connection with the theory of partial differential equations of elliptic-parabolic type (possibly degenerate) and of the first order. They will be able to apply the acquired knowledge to solve, even numerically, various types of problems concerning some classical kinetic models of physics and the theory of stochastic processes.

Course contents

Probability spaces. Dynkin's Theorems. Distributions. Random variables. Integration. Expectation. Sigma-algebras and information. Independence. Fourier transform. Cauchy problem for second order parabolic partial differential equations with constant coefficients. Fundamental solution of the heat equation. Existence and uniqueness of the solution of the Cauchy problem for the heat operator. Multinormal distribution. Characteristic function. Conditional expectation. Conditional expectation. Discrete stochastic process. Martingales. Discrete stochastic integral. Stopping time and Optional sampling theorem. Markov inequalities. Doob's maximal inequality for discrete martingales. Stochastic processes. Brownian motion. Brownian motion: property. Brownian martingales. Doob's maximal inequality. Exponential distribution. Poisson process, compound and plywood. Law of a continuous process. Amendments and processes indistinguishable. Distributions finite-dimensional Brownian motion. Functions to change limintata Riemann-Stieltjes integral and the Poisson process. Deterministic Ito formula. Quadratic variation of the Brownian motion. Wiener's stochastic integral. Ito's integral of simple processes. Usual hypotheses on filtrations. Stopping times and martingales. Integral in L ^ 2. Properties of the integral of a L ^ 2 process. Ito integral and stopping times. Quadratic variation process. Martingale bounded variation. Integral of a L ^ 2_loc process. Local martingales. Ito process. Ito formula for the Brownian motion. Multidimensional Brownian motion. General Ito formula. Examples. Stochastic differential equations. Existence and uniqueness. Characteristic operator. Exit time of a diffusion from a bounded domain. Representation formulas for PDE problems related to SDEs. Examples. Method of characteristics. Cauchy-Dirichlet problem for parabolic equations. Fundamental solution and transition density. Monte Carlo method and numerical solution of the Dirichlet problem for ordinary differential equations second-rate. Linear SDEs and control problems. Condition of Kalman and controllability.

Readings/Bibliography

A. Pascucci, PDE and Martingale methods in option pricing, Bocconi &  Springer Series, 2010

http://math-finance.blogspot.com/

Teaching methods

Lectures.

Assessment methods

An oral examination will be made, properly testing the student on the knowledge of the course
contents. It is left to the choice of at least one argument. It is normally required the proof or at least trace of the proof of the main results.

Links to further information

https://docs.google.com/document/d/13IZPw0JlHuzmgZ2p0bXsg8MXum1RlNiS_GYEtbBNKNE/edit?usp=sharing

Office hours

See the website of Andrea Pascucci