FSF3713 Stochastic Analysis 7.5 credits

• Stochastic processes, martingales, local martingales, stopping times, filtrations, Markov properties.
• Brownian motion.
• The Ito isometry, Ito integrals, Ito’s formula.
• Existence and uniqueness of solutions to stochastic differential equations.
• Diffusion processes.
• Girsanov’s theorem.
• Probabilistic representations of solutions to partial differential equations.
• The Feynman­Kac formula, Kolmogorov’s forward and backward equations, recurrence, invariant densities.
• More advanced topics, e.g., local times, if time permits.

• Understand and explain the following concepts: filtration, stochastic process in continuous time, local martingale, martingale, stopping time, quadratic variation.
• Sketch at least one construction of Brownian motion.
• Construct the Ito integral in some generality, and discuss its basic properties.
• Explain Ito’s formula and use it for practical computations of, e.g., Ito integrals.
• Discuss basic properties of stochastic differential equations (SDEs), in particular diffusions in one dimension.
• Explain Girsanov’s theorem.
• Discuss connections between the theory of SDEs and partial differential equations.
• Solve problems and discuss current research connected to the theory presented in the course

Basic probability theory (e.g., SF3940) and basic knowledge in analysis and linear algebra, especially measure theory and Lebesgue integration.

For example

Karatzas­Shreve “Brownian motion and stochastic calculus” (ISBN 978-1-4612-­0949-­2),

Revuz­Yor “Continuous martingales and Brownian motion” (ISBN 978­-3-662­-06400-­9);

Öksendal “Stochastic Differential Equations” (ISBN 978-­3-­642­14394­-6)

If the course is discontinued, students may request to be examined during the following two academic years.

• INL1 - Assignment, 3.5 credits, grading scale: P, F
• TENM - Oral exam, 4.0 credits, grading scale: P, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

Homework assignments and oral/written exam.

Homework assignments completed and oral/written exam passed.

Fredrik Viklund

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.

PhD students only.