Textbook: Stochastic Processes by Rich Bass -- (also available from amazon.com) -- (see also lecture notes)
Syllabus: Continuous time random processes, Kolmogorov's continuity theorem. Brownian Motion: the Donsker invariance principle, Holder continuity, quadratic variation. Continuous-time martingales and square integrable martingales. Markov processes and the strong Markov property. Properties of Brownian Motion: strong Markov property, Blumenthal zero-one law, Law of Iterated Logarithm. Stochastic integration with respect to continuous local martingales, Ito's formula, Levy's Characterization of Brownian motion, Girsanov transformation, Stochastic Differential Equations. Other topics will be included, depending on the interest of students. No written homework will be collected, but the course will include short presentations by students of topics of mutual interest related to stochastic analysis.
Schedule of the course:
HW due Monday, January 22: read Section 1, including exercises.
HW due Wednesday, January 24: read Section 2, including exercises.
HW due Monday, January 29: read Section 3, including exercises.
HW due Wednesday, January 31: read Section 4, including exercises.
HW due Friday, February 2: read Section 5, including exercises.
HW due Monday, February 12: read Sections 6,7,8, including exercises.
HW due Monday, February 19: read Sections 9,10, including exercises.
HW due Monday, February 26: read Sections 11,12, including exercises.
March (after the break): Sections 13, 24, 18, 17
April: 42, 17, 19-25, tba
Topics of discussion: Semigroups, Dirichlet forms and symmetric Markov processes, Sections 36-40
Stratonovich integral and rough path properties
Last class meeting: Wednesday May 2, 6pm-8pm
Applications to financial math, sections 28, 35.2