# MAI0042

Continuous stochastic calculus

**Number of credits:** 12 hp

**Examiner:** Timo Koski, John M. Noble

**Course literature:** Bernt Øksendal, *Stochastic Differential Equations : An Introduction with Applications* (latest edition). Springer.

**Course contents:**

1. **Itô Integrals**

The basic Itô integral driven by a Brownian motion. The increasing process for continuous time martingales. The martingale representation theorem for continuous martingales.

2. **Stochastic Differential Equations** Examples, definition, notions of solution: weak and strong, existence problems, conditions for uniqueness of solution.

3. **Filtering** One dimensional and multidimensional filtering problems. The Kalman Bucy filter.

4. **Diffusions** The Markov property, the infinitesimal generator, Kolmogorov's backward equation, the resolvent, the Feynman Kacs formula, Girsanov's theorem, random time change, application to boundary value problems.

5. **Application to Optimal Stopping and Stochastic Control**

Time homogeneous and time inhomogeneous optimal stopping problems, the Hamilton Jacobi Bellman equation, stochastic control with terminal conditions.

6. **Application to Mathematical Finance** Market, Portfolio and Arbitrage, Attainability and Completeness, Option Pricing and the Black Scholes formula.

**Organisation:** The course contains 12 lectures.

**Examination:** Examination is by hand in assignments, one following each lecture.

**Prerequisites:**

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Last updated: 2014-04-29