# MA682 Stochastic Analysis

**Lecturer: Vassili Gelfreich**

**Term(s):** Term 2

**Commitment:** 30 lectures

**Assessment:** Oral exam

**Assumed knowledge:**

- Basic ideas of Probability Theory as in ST111 Probability (Part A) or ST112 Probability (Part B): Random variables, expectations, mean and variance, central limit theorem, law of large numbers.
- Some experience of stochastic processes. In past years about 80% of students had taken MA6F7/ST403 Brownian Motion (although this module will recap Brownian motion and only a few properties are needed). The modules ST202 Stochastic Processes or ST333 Applied Stochastic Processes would be valid alternatives.
- Measure Theory: This module will use the key weapons of rigorous measure theory (measurable functions, integrals, Fubini's Theorem, Dominated Convergence Theorem, Fatou's lemma) as seen in MA359 Measure Theory or ST342 Mathematics of Random Events. The module gives a chance to see these ideas in action, but it will not stress measure theoretic aspects.

**Useful background:** There will be links with material from several other modules: Solutions to elliptic and parabolic linear PDEs are very closely related, and students who have taken MA250 Introduction to Partial Differential Equations, MA3G1 Theory of Partial Differential Equations or MA6A2 Advanced Partial Differential Equations will be motivated for some of these connections. Although we won't lean on the material developed in ST318 Probability Theory, the examples developed here motivated a lot of the theory of martingales.

**Synergies: **Financial mathematicians make use of the tools developed here so that it will eventually mesh with the ideas in ST339 Introduction to Mathematical Finance (perhaps in Masters level courses).

**Leads to:**

**Content**: We will introduce stochastic integration, and basic tools in stochastic analysis including Ito’s formula. We will also introduce lots of examples of stochastic differential equations.

**Books**:

Laurence Evans: *An Introduction to Stochastic Differential Equations*.

Bernt Oksendall: *Stochastic Differential Equations*.