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MAI0095
Stochastic Calculus/
Stokastisk analys

Number of credits: 8 hp

Examiner: John Noble

Course literature: Bernt Oksendal: Stochastic Differential Equations: An Introduction with Applications (6th Edition) Springer (2003). Unfortunately, there are serious omissions in a substantial number of the proofs. A course compendium, containing statements of the theorems together with their proofs and very little motivation can be found on the course home page.

Course contents:

• Foundational material for continuous time processes: review of measure theory, review of conditional expectation, review of Kolmogorov's continuity criterion (these topics are treated fully in the ''Graduate Course in Probability Theory''), Kolmogorov's extension theorem, regular conditional probability distributions, filtrations and stopping times.
• Review of the Wiener process and Gaussian processes (these topics are treated more fully in the ''Graduate Course in Probability Theory'').
• Continuous martingales.
• The Doob-Meyer decomposition, quadratic variation, stochastic integrals, Ito's formula and first applications, the Burkholder-Davis-Gundy inequality.
• Local times: definition, the Tanaka formula, continuity properties, specific properties of the Local time of the Wiener process.
• Stochastic differential equations: formal definitions, pathwise uniqueness, uniqueness in law, strong solutions, weak solutions, existence and uniqueness results with Lipschitz coefficients.
• The Kalman - Bucy filter.
• Feller processes, strong Markov property, diffusions, infinitesimal generators and Itô processes, the martingale problem.
• Representations of solutions and probabilistic methods for partial differential equations: The Feynman Kacs representation, Poisson and Dirichlet problems.
• The martingale representation theorem and Girsanov's theorem.

Organisation: 12 lectures, two per week, in HT1.

Examination: Six substantial written assignments, one per week, for the duration of the course.

Prerequisites: The ''Graduate course in probability'' based on the book by R. Durrett, or a course in measure theory, is strongly recommended.

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Last updated: 2014-04-29