# ST908: Probability & Stochastic Processes

###### Lecturer(s)

###### Prof David Hobson

** Commitment: **1 x 3 hour lectures + 1 seminar per week. This module runs in Term 1.

**Important: This module is Core for MSc Financial Mathematics students. It is not available to undergraduates. Other students interested in taking the module should consult the lecturer.**

*Content***: **

1. Sample spaces, events and probabilities. Conditional probability. Independence. Discrete random variables including geometric, Poisson. Expectation.

2. Continuous random variables, including Gaussian, exponential, gamma. Distributions of functions of functions of random variables. Sums of random variables. Conditional distributions.

3. Borel-Cantelli Lemmas, Convergence of random variables. Convergence of expectations.Fatou's Lemma.

4. Weak law, strong law. Connections with Monte-Carlo simulation. Central limit theorem.

5. Filtrations, information conditional expectation. Martingales in discrete and continuous time. Stopping times.

6. Stochastic processes. Markov property. Simple random walks. Ruin probabilities. Poisson process.

7. Brownian motion. Quadratic variation, volatility. Hitting times for linear boundaries. Simple models for bankruptcy.

8. Ito’s formula. Stochastic integrals. Gains from trade processes. Stochastic differential equations.

9. Diffusions, exponential Brownian motion. Samuelson model for stock prices. Scale functions.

10. Brownian martingale representation theorem, Girsanov change of measure. Black- Scholes formula for call options.

*Texts***: **

Williams D., (1991) Probability with Martingales, Cambridge CUP.

Grimmett G. and Stirzacker D., (2001), *Probability and Random Processes*, 3^{rd} edition, Oxford: OUP.

Oksendal B., (2003), *Stochastic Differential Equations: An Introduction with Applications*, 6^{th} edition, Berlin and Heidelberg: Springer-Verlag.

Shreve, S.E., (2005), *Stochastic Calculus for Finance*, New York: Springer-Verlag.

*Assessment***: Examination (80%), Coursework (20%)**

Resources for Current ST908 Students (restricted access)