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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