Math 230C / Stat 310C : Syllabus

Construction of stochastic processes from finite-dimensional distributions. Continuous modifications. Kolmogorov-Centsov theorem.

(8.1, 8.2)

Handwritten notes: Lecture 1, Lecture 2

Relation with Borel sigma algebra of continuous function. Separability. Measurable stochastic processes. Gaussian processes. Definition of Brownian motion.

(8.2, 8.3)

Handwritten notes: Lecture 3, Lecture 4

Filtrations and adaptedness. Stopping times and Markov times: basic properties. Progressive measurability. Relation with hitting times. Doob's maximal inequality. Doob's martingale convergence theorem. Uniform integrability and convergence in ell one.

(9.1, 9.2.1, 9.2.2)

Handwritten notes: Lecture 5, Lecture 6

Optional stopping theorem. Markov processes: defitions and simple Markpov property

(9.2.3, 9.3.1)

Handwritten notes: Lecture 7, Lecture 8

Strong Markov property. Feller semigroups. Homogeneous Markov processes with right continuous sample paths and Feller semigroup are strong Markov.

(9.3.2)

Handwritten notes: Lecture 9, Lecture 10

Brownian motion. Symmetry and strong Markov property. Consequences, reflection principle. Donsker's invariance principle.

(10.2.1)

Handwritten notes: Lecture 11, Lecture 12

Skorokhod's embedding. Strassen's theorem. Martingale Central Limiti Theorem.

(10.2.1)

Handwritten notes: Lecture 13, Lecture 14

Regularity properties of the Brownian motion. Law of the iteratedlogarithm. Nowhere differentiability.

(10.2.2, 10.3)

Handwritten notes: Lecture 15, Lecture 16

Stochastic integral with respect to Brownian motion. Ito's formula.

(Morters-Peres, 7.1)

Handwritten notes: Lecture 17, Lecture 18

Conformal invariance of Brownian motion. Feynman-Kac formula.

(Morters-Peres, 7.2, 7.4)