Slides & board photos, material covered

Board photos & synopsis of material covered & links to lecture notes:

lecture 1, January 8: Intro to the course. Definition of a stochastic process (generally valued). Examples: homogeneous Poisson process and standard Brownian motion (Notes & slides)
lecture 2, January 10: Finite dimensional distributions of a stochastic process and their consistency. Kolmogorov Extension Theorem. The construction of a general product measure space. The standard Borel spaces and inner regularity of probability measures (Notes & Photos)
lecture 3, January 12: Proving subadditivity of premeasure induced by consistent family of finite-dimensional distributions. Using Hahn-Kolmogorov to prove Kolmogorov Extension Theorem. Uniqueness of the extension via Dynkin's π/λ-Theorem. Non-measurability in the product space of the set of continuous paths; countability curse. A version/modification of a process (Notes & Photos)
lecture 4, January 17: Statement and proof of Kolmogorov-Čenstov Theorem. Failure of KČ-continuity condition for the homogeneous Poisson process (which is not continuous). Existence of Brownian motion by checking KČ-condition from finite-dimensional distributions of Browninan motion. (Notes & Photos)
lecture 5, January 19: A process with continuous path can be thought of as continuous-function valued random variable. Wiener space and Wiener measure. Brownian symmetries: shift, diffusive scaling and time inversion. SLLN for standard Brownian motion. Sample path regularity: The Paley-Wiener-Zygmund Theorem on non-differentiability, and non γ-Hölder continuity for γ<1/2 of Brownian paths. Proof next time. (Notes & Photos)
lecture 6, January 22: Proof of Paley-Wiener-Zygmund nowhere differentiability. Review of detailed path regularity: Khinchin's (1933) Law of the Iterated Logarithm, Levy's (1937) modulus of continuity and Brownian fast points, existence of 1/2-Hölder points (Dvoretzky 1963, Davis 1983) and Brownian slow points. Calculations motivating Itô calculus: Convergence of second variation of a Brownian path as mesh of the partition tends to zero. Discovering Itô integral in the derivation of Itô's formula. (Notes & Photos)
lecture 7, January 24: Existence of (Itô) stochastic integral ∫₀^t f(B_s) dB_s for functions f with bounded derivatives and then just bounded functions. Itô formula. Stratonovich integral and its pros and cons. Motion equation with a noise term in differential and integral form. (Photos)
lecture 8, January 26: Two examples of noise processes with independent increments: Poisson point process and white noise. Itô isometry for the white noise and the Paley-Wiener integral. (Notes and Photos)
lecture 9, January 29: Stochastic integral introduced in full generality. Brownian filtration, simple process and stochastic integral thereof. Itô isometry and extension of the stochastic integral to the closure of the space of simple processes. Inclusion of left continuous processes therein. Computing stochastic integral by Itô formula or integration by parts in special cases. (Notes and Photos)
lecture 10, January 31: Continuing the construction of the stochastic integral: existence of a continuous version of the stochastic integral (which is also an L²-martingale). Simple processes are dense in the set of jointly measurable, adapted processes that are square integrable on compact intervals of time. (Notes and Photos)
lecture 11, February 2: Finishing a proof of approximation lemma from last time. Using the localization technique to extend the Itô integral to processes that are just locally square integrable with respect to time a.s. Proofs to be finished next time. (Notes and Photos)
lecture 12, February 5: Details of the extension of the Itô integral to locally square integrable processes. The key role of the identity ∫₀^{t^T} Y_sdB_s = ∫₀^t Y_s1_{T>s} dB_s for any stopping time T. Discretization of a stopping time. (Photos)
lecture 13, February 7: Independence of localization of the sequence of stopping times. Linearity of integral in underlying process, additivity under partition of the integration domain. Itô integral of a process determines the process up to null-set modifications. Convergence of (left-endpoint) Riemann sums to stochastic integral for continuous integrands. (Notes and Photos)
lecture 14, February 9: Itô formula. Stratonovich integral from Itô integral. The Wong-Zakai Theorem showing that integrals with respect to smooth version of Brownian motion converge to the Stratonovich integral. The notion of a generalized diffusion a.k.a. continuous semimartingale as a process given by a sum of an Itô integral and an ordinary integral with respect to time. Itô diffusion as a special case thereof. Differential form: dX_t=Y_tdB_t+U_tdt. Quadratic variation of a continuous semimartingale. (Notes and Photos)
lecture 15, February 12: Proof that quadratic variation of stochastic integral ∫₀^t Y_sdB_s is the process ∫₀^t Y_s²ds. Quadratic variation as a compensator process making square of stochastic integral a martingale. Relation to, and uniqueness in, Doob-Meyer decomposition. Itô formula for semimartingales. (Photos)
lecture 16, February 14: Itô formula for C² function of a univariate semimartingales. Proof given in full detail. (Notes and Photos)
lecture 17, February 16: Extensions of Itô formula to multivariate semimartingales. Application to exponential martingale arising from a stochastic integral and radial variable of d-dimension standard Brownian motion. The product rule and integration by parts. The Fisk-Stratonovich integral for semimartingales. Levy characterization of Brownian motion (Levy 1949): a continuous local martingales whose quadratic variation is linear in time is a Brownian motion. (Notes and Photos)
lecture 18, February 21: Application of Levy characterization: The radial variable of d-dimensional standard Brownian motion satisfies Bessel's SDE. Facts from continuous-martingale theory: stopping times, Optional Stopping/Sampling Theorem. Time change that turns a stochastic integral into a standard Brownian motion (proof given next time). (Notes and Photos)
lecture 19, February 23: Proof that time change of a stochastic integral by inverse of its quadratic variation is standard Brownian motion. The definition of d-dimensional Bessel process. A natural martingale associated with this process. (Notes and Photos)
lecture 20, February 26: The martingale associated with Bessel process and relation to hitting times of level sets. The phases of the Bessel process and the connection with recurrence/transience of d-dimensional standard Brownian motion. (Photos)
lecture 21, February 28: Solving Dirichlet problem for Laplace equation in bounded open subsets of Euclidean space. Representation of continuous solutions via exit problem for Brownian motion. Multiplicity of solution under positive survival probability for Brownian motion. (Notes and Photos)
lecture 22, March 1: Strong Markov property and Blumenthal's Zero-One Law proved as part of technical arguments. Boundary behavior of the probabilistic solution to Dirichlet-Laplace problem. The concept of a regular point as that from which a.e. Brownian path exists the domain instantaneously. Equivalence of regularity and continuity of every bounded solution. (Notes an Photos)
lecture 23, March 4: Proof of a technical proposition underpinning the characterization of regular boundary points. Other PDEs that can be represented via expectations with respect to standard Brownian motion: Poisson equation, Helmholtz equation, heat equation. A representation for eigenfunction/eigenvalue proglem. The Feynman-Kac formula. (Notes and Photos)
lecture 24, March 6: Stochastic differential equations. Introduction and motivation, definition of "standard setting" and the concept of a strong solution. Existence of a strong solution for SDEs with uniformly Lipschitz coefficients. (Notes and Photos)
lecture 25, March 8: A solution of an SDE up to a stopping time. Uniqueness of such solutions under uniform Lipchitz property of the coefficients. Existence and uniqueness of the strong solution under locally Lipschitz conditions. Existence of Bessel processes. Non-existence of a strong solution for the Tanaka equation. (Notes and Photos)
lecture 26, March 11: The concept of pathwise uniqueness. The concept of a weak solution. Failure of pathwise uniqueness for weak solutions to the Tanaka equation. (Notes and Photos)
lecture 27, March 13: Uniquness in law. Yamada-Watanabe "abstract-nonsense" theory (pathwise uniqueness implies uniqueness in law and that pathwise uniqueness and weak existence imply strong existence, uniformizing map onto a canonical solution space, the solution map under existence and pathwise uniqueness). Only highlights given; details to be found in Notes. Methods for solving SDEs: Reduction to ODE for SDEs with unit dispersion, removal of drift via change of coordinates. Time change in stochastic integral. (Notes and Photos)
lecture 28, March 15: Application of time change to solution of SDEs with vanishing drift term under simple integrability criterion. Continuing methods for solving SDEs: Proof that dX_t = σ(X_t) dB_t admits a weak solution whenever 1/σ² is locally Lebesgue integrable. Statement of the 1971 Engelbert-Schmidt Theorem giving necessary and sufficient conditions for existence and uniqueness in this case. The method of measure change. Girsanov's Theorem. (Notes and Photos)