Course Objectives

1. Introducing Stochastic Calculus and its rules.

2. Introducing Stochastic Differential Equations.

3. Showing their utility in modelling evolutions with noise in different contexts such as Finance, Engineering, Chemistry, Physics.

Course Content

1. Elements of Probability. Probability spaces, random variables. Variance, covariance, probability distribution, density. Independence. Random vectors. Convergence of random variables. Characteristic functions. Gaussian laws. Measurability theorems.

2. Stochastic processes. Filtrations. Trajectories. Equivalent processes, modifications, indistinguishable processes. Finite-dimensional laws and Existence Kolmogorov’s theorem. Kolmogorov’s continuity theorem. Stopping times.

3. Brownian motion. Definition and basic properties. Finite-dimensional distributions. The White Noise.

4. Conditional probability. Conditional expectations. The augmented Brownian filtration.

5. Martingales. Definitions and basic properties of continuous time Martingales.

6. The stochastic integral. Elementary processes. The stochastic integral. Ito Isometry. The stochastic integral as a process. Stopping theorems. Local martingales.

7. Stochastic calculus. Stochastic differential of an Ito process. Ito's Lemma. Girsanov’s Theorem. The martingales of the Brownian filtration.

8. Stochastic Differential Equations. A class of SDE. Definition of solutions. Existence and Uniqueness theorems for the solution. SDE and Markov processes. Connections between SDE and PDE. Feynman-Kac formula.

The course final exam is made of a preliminary written test, followed by an oral test, whose access is subject to a passing rate in the written test.

Draft of a forthcoming book, courtesy of the Author.