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

• lecture 1, April 3: Intro to the course. Review of basic topics: stochastic processes, measurability, filtrations. Martingales, convergence results, inequalities. Stopping times and the Optional Stopping/Sampling Theorem. (Notes & Photos --- two boards unfortunately missing)
• lecture 2, April 4: Continuous local martingales admit a quadratic variation process. Construction as the limit of second variations (Notes & Photos)
• lecture 3, April 5: Itô integral of simple processes and its extension to the closure of simple processes induced by pseudometric associated with a continuous square-integrable martingale. (Notes & Photos)
• lecture 4, April 8: Characterizing Itô-integrable processes with respect ot continuous martingales whose sample paths are, and are not, absolutely continuous. The role of progressive measurability in the latter case. See also Ondreját-Seidler's paper. (Notes & Photos)
• lecture 5, April 10: Localized version of Itô integral. Approximation of integrands in probability approximates integrals in probability. Continuous semimartingales and Itô formula. Levy characterization of Brownian motion. Writing a continuous local martingale with AC quadratic variation as Itô integral. (Notes & Photos)
• lecture 6, April 11: Representation of continuous martingales as Itô integrals w.r.t. standard Brownian motion. The case of Brownian motion derived from the martingale under assumption that its quadratic variation is absolutely continuous and then a martingale measurable w.r.t. a Brownian filtration. (Notes & Photos)
• lecture 7, April 12: Finishing Brownian martingales. Exponential change of measure/tilting for independent random variables and multivariate Gaussian. Discrete-time and continuous time Girsanov theorems. (Notes & Photos)
• lecture 8, April 15: Girsanov's theorem. Characterization of when an exponential supermartingale is a martingale. Example of a local (exponential) martingale that is not a proper martingale. Sufficient conditions to get a martingale. (Notes & Photos)
• lecture 9, April 17: Application of Girsanov's theorem: removal of a drift from Lengevin type SDE. Construction of standard Brownian motion conditioned to stay positive by a limit procedure and identification of the limit as a 3-dimensional Bessel process. (Notes & Photos)
• lecture 10, April 18: Necessary and sufficient conditions for the law of a Brownian path shifted by a deterministic function to be absolutely continuous with respect to the Wiener measure. (Notes & Photos)
• lecture 11, April 19: Finishing a proof from the last lecture; the Cameron-Martin theorem. Gaussian Hilbert spaces. The Cameron-Martin space (after Cameron and Martin 1944). Examples for stochastic integral with respect to Brownian motion and white noise. (Notes & Photos --- unfortunately, the board with definitions and examples of GHS missing.)
• lecture 12, April 22: The Karhunen-Loève expansion of Gaussian processes. Uniformity of convergence in L² for continuous processes over compact index sets. Uniformity of convergence in supermum metric with the help of Itô-Nisio theorem. (Notes & Photos)
• lecture 13, April 24: The Cameron-Martin theorem for general mean-zero Gaussian processes; proof using exhaustion of the index set by finite sets. (Notes & Photos)
• lecture 14, April 25: Wiener chaos expansion (proposed in Wiener 1938). Orthogonality to all polynomials in variables of Gaussian process implies triviality. A representation of L² space associated with a Gaussian process an infinite direct sum of Wiener chaos spaces. Examples of a single Gaussian random variable and the white noise. (Notes & Photos)
• lecture 15, April 26: Wick normal ordering of products of elements from a Gaussian Hilbert space. Key facts about this concept. Introduction of higher-order chaos integrals (Notes & Photos)
• lecture 16, April 29: Construction of higher-order chaos integrals. Properties of the Wick product. Wick-ordered exponential. (Notes & Photos)
• lecture 17, May 6: Iterated Itô integrals. Itô chaos decomposition. Relation to Brownian martingales. (Notes & Photos)
• lecture 18, May 8: Solving the Stochastic Heat Equation (SHE). Mollified noise, integral representation. Removal of smoothing in dimension one. Holder regularity. (Notes & Photos)
• lecture 19, May 9: The solution constructed for one-dimensional SHE is a weak solution. The KPZ equation in Cole-Hopf form. (Photos)
• lecture 20, May 10: Solving the one-dimensional KPZ in Cole-Hopf form by expansion in iterated Paley-Wiener integrals. (Notes & Photos)
• lecture 21, May 13: Towards an exponential form of Cole-Hopf solution. Feynman-Kac formula for the solution of heat-equation with multiplicative force term (Notes & Photos)
• lecture 22, May 15: The Feynman-Kac representation for the Cole-Hopf solution of one-dimensional KPZ equation. (Notes & Photos)
• lecture 23, May 30: The Young integral or, more precisely, the Young criterion for Stieltjes integrability. The Love-Young inequality. (Notes & Photos)
• lecture 24, May 31: Finishing the proof of the Young criterion. Steps towards solutions of moderately rough ODEs. Functions of finite p-variation and the associated Banach spaces. Examples: Weierstrass function, iterated standard Brownian motions, Fractional Brownian motion. (Photos)
• lecture 25, June 3: Framework for solving moderately rough ODEs. The Banach space of finite p-variation. Invariance under reparametrization, relation to Hölder continuity. Statement of Peano's theorem. (Notes & Photos)
• lecture 26, June 5: Proof of generalized Peano theorem on existence of solutions of ODEs driven by signals of moderate roughness. The role of Hölder continuity. Statement of Picard's theorem and proof of a key inequality. (Notes & Photos)
• lecture 27, June 6: Proof of Picard's theorem. Lack of continuity of the Stieljes integral with respect to 2-variation norm. Motivations for the consideration of rough paths: the role of iterated integrals, observation that iterating integrals improves regularity. (Notes & Photos)
• lecture 28, June 7: A quick overview of rough path theory. The concept of signature of a smooth path. Chen's identity and its generalization to multiplicative functionals of finite order. Functionals of finite p-variation and existence and uniqueness of their extension. A preview of what lies beyond ... (Notes & Photos)