Slides & board photos, material covered

Lecture notes: (WARNING: These can change day to day.)

Slides & board photos & synopsis of material covered:

Lecture 1, Sept 26: Introduction to the course. A stochastic process as a family of random variables (in the Kolmogorov model). Key examples: Homogeneous Poisson process and standard Brownian motion. A quick note on Lévy processes. Finite dimensional distributions and their consistency. A statement of the Kolmogorov Extension Theorem. (photos -- unfortunately incomplete)
Lecture 2, Sept 29: Statement and proof of the Kolmogorov Extension Theorem. Arbitrary-index product spaces and product σ-algebras. Inner regularity of finite Borel measures on standard Borel spaces. Key fact for measure extension --- namely, countable subaddivity of P --- proved by showing that B_n↓∅ implies P(B_n)↓0 via the Cantor intersection property. Uniqueness via Dynkin's π/λ-Theorem. (photos)
Lecture 3, Oct 1: Continuity of sample paths cannot be determined from finite-dimensional distribution. Non-measurability of the set of continuous paths in product-space setting. A version/modification of a process. Toward criteria for existence of a continuous version: stochastic continuity necessary but not sufficient. The Kolmogorov-Čenstov Theorem. Proof of existence of standard Brownian motion and γ-Hölder continuity of its sample paths, for all γ<1/2. (photos)
Lecture 4, Oct 2: A process whose every sample path is continuous defines a function-space valued random variable. The law of this random variable is in turn determined by finite-dimensional distributions. A consequence: uniqueness of Wiener measure on the canonical space. Sample path properties: Invariance under time shift, diffusive scaling and time inversion. SLLN for standard Brownian motion. Symmetries in higher dimensions. (photos)
Lecture 5, Oct 3: Statement and proof of a theorem of Paley, Wiener and Zygmund from 1933 on nowhere differentiability (and nowhere γ-Hölder continuity, for γ>1/2) of a.e. Brownian path. Refinements (just statements): Khinchin's (1933) Law of the Iterated Logarithm, Lévy's (1937) modulus of continuity and Brownian fast points, Dvoretzky's (1963) and Davis' (1983) theorems on Brownian slow points. Quadratic variation of Brownian paths and differentiable functions thereof. (photos)
Lecture 6, Oct 6: Discovering the Itô integral by representing increments of a function along Brownian path to second order in Taylor theorem with integral remainder. Extension to all continuous functions of the Brownian path. Itô formula and the lack of proper Riemann-Stieltjes integrability. Stratonovich integral via mid-point rule. (photos)
Lecture 7, Oct 8: Interpreting ODEs with a noise term. Treating noise as a random signed measure. Poisson process from first lecture arises from a Poisson point process on positive reals and the Lebesgue measure. Compensated and compound Poisson random variable. Gaussian white noise and its different characterizations. The Paley-Wiener integral as a linear map between L²-spaces. (photos)
Lecture 8, Oct 20: Definition of Itô stochastic integral of general processes w.r.t. Brownian motion. The notions of a Brownian filtration and a simple process. The integral of a simple process and Itô isometry. Extension of the integral to L²-closure of the space of simple processes. Left-continuous adapted square integrable processes are integrable. (photos)
Lecture 9, Oct 22: Itô stochastic integral as a continuous square-integrable martingale. Key tool: Doob's L²-inequality. Characterizing the class of integrable processes as those that are jointly measurable, adapted (to a Brownian filtration) and locally square integrable. (photos)
Lecture 10, Oct 24: Further extension of Itô stochastic integral to processes that a jointly measurable, adapted but only locally square integrable in time, for every compact interval of times a.s. Key tool: localization. The notion of stopping time and its discretization. The bulk of the proof is devoted to the identity ∫₀^T∧t Y_sdB_s = ∫₀^t Y_s1_{T>s}dB_s for all (square integrable, jointly measurable and adapted) processes Y and all stopping times T, where the left integral is obtained by plugging T∧t for time in a continuous version of the stochastic integral. The identity is shown explicitly for Y simple and discretized T; taking limits then extends this to full generality.(photos)
Lecture 11, Oct 27: Proof of existence of Itô integral for locally integrable processes. Coincidence of this notion with integral defined via Itô isometry, independence of stopping time used for localization. Necessity of local square integrability. Additivity of the integral w.r.t. integrand and integration domain. Conditional Itô isometry, the fact that {(∫₀^t Y_sdB_s)^2-∫₀^t Y_s²dB_s: t≥0} is a martingale. Identification of the integrand from Itô integral. (photos)
Lecture 12, Oct 29: Review of properties of Itô integral introduced last time. Characterizing integrable processes as in probability limits in L² with respect to time. Generalized diffusions as sums of ordinary integrals and Itô integrals. Infinitesimal form dX_t=U_t dt+Y_tdB_t. Itô formula for generalized diffusions. Proof for diffusions given via simple processes. (photos)
Lecture 13, Oct 31: Proof of Itô formula for diffusions; the part where the claim is extended from simple processes to general processes. An extension (without proof) to diffusions driven by multiple Brownian motions. The mnemonic rules: (dt)²=0, (dt)(dB_t) = 0, (dB_t⁽ⁱ⁾)(dB_t^(j))=δ_i,j dt whenever B⁽¹⁾,..., B^(d) are independent standard Brownian motions. Martingales and local martingales. The quadratic variation process. (photos)
Lecture 14, Nov 3: The quadratic variation of a continuous local martingale. Proof of uniqueness (which already gives the statement for local martingales arising from Itô integrals). Uniqueness of representation of diffusions. A key lemma for the proof of existence. Proof to be finished next time. (photos)
Lecture 15, Nov 5: Finishing proof of existence of quadratic variation of a continuous local martingale. Stochastic integral ∫₀^t Y_sdM_s of simple Y with respect to continuous L²-martingales M: Itô isometry, the fact that {(∫₀^t Y_sdM_s)^2-∫₀^t Y_s²dM_s: t≥0} is a martingale for simple Y. Extension to closure of simple processes including existence of continuous version. (photos)
Lecture 16, Nov 6: Addressing the question which processes Y can be integrated with respect to a given continuous L²-martingale. Reduction to the previous case for t↦<M>_t absolutely continuous a.s. Progressive measurability and its relation to X_T being measurable for T a stopping time. Integrability of (only) progressively measurable processes. (photos)
Lecture 17, Nov 7: Extension of stochastic integral to locally integrable processes. Continuity in integrand and integrator. Continuous semimartingales, covariation. Itô formula for functions of continuous semimartingales. Product rule/integration by parts formula. Formal recovery of both FTC and IBP in terms of Fisk-Stratonovich integral. Multivariate Itô formula. (photos)
Lecture 18, Nov 10: Lévy characterization of standard Brownian motion as a continuous local martingale M with <M>_t=t for all t. Time change to standard Brownian motion for continuous local martingales with <M> strictly increasing and diverging to infinity. Representation as Itô integral for continuous local martingales with absolutely continuous <M>. (photos)
Lecture 19, Nov 12: Proof of the representation of continuous local martingales with absolutely continuous <M> as Itô integral with respect to a standard Brownian motion. Key input: substitution rule for stochastic integrals. The statement and proof that, for all X square integrable on a space supporting a standard Brownian motion B, there exists an integrable process Y such that E(X| B_s: s ≤ t) = EX+∫₀^t Y_sdB_s a.s. for all t≥0. (photos)
Lecture 20, Nov 13: Finishing the proof of the representation theorem from last time. The corresponding representation of a continuous L²-martingale. Iterated Itô integrals built from multivariate simple processes. (photos)
Lecture 21, Nov 14: Representation of the space of square integrable function of a path of Brownian motion as an infinite sum of iterated Itô integrals. Another proof of the representation theorem from last time. Itô formula casting the iterated integral of a constant as the Hermite polynomial of the value of Brownian motion. (photos)
Lecture 22, Nov 17: The Bessel process as radial process for d-dimensional standard Brownian motion. Definition for all real valued d. Probability of a diffusion hitting a level set; approach via harmonic function/reduction to martingale. Explicit form of the function for Bessel process. Statement of the main theorem; some details from proof to be given next time. (photos)
Lecture 23, Nov 19: Finishing Bessel processes: zero hit a.s. in d<2 and not in d≥2, yet all positive levels hit in d=2 while the process remains bounded away from zero when d>2. Consequences for d-dimensional standard Brownian motion: recurrence to points in d=1, recurrence to open balls but transience to points in d=2, transience to open balls in d≥3. Stochastic differential equations, the notion of the "standard setting" and a strong solution. Itô's theorem on existence of a strong solution under Lipschitz continuity of the coefficients. (photos)
Lecture 24, Nov 20: Proof of Itô's theorem on existence of a strong solution under Lipschitz continuity of the coefficients. Checking that square integrability of initial value propagates through Picard-Lindelöf iterations. Iterative bound on difference of the iterates squared, uniformly over compact intervals of time and under expectation. Construction of the solution. The concept of strong solution up to a stopping time. Statement of uniqueness and locality of solution up to a stopping time under the Liptschitz condition on the coefficients. (photos)
Lecture 25, Nov 21: Proof of uniqueness & locality theorem under Lipschitz continuity of the coefficients. Reduction to Gronwall's lemma. Exhausing domain of local Lipschitz continuity. Pathwise uniqueness. Existence and pathwise uniquencess of strong solution of Bessel equation (and of d-dimensional Bessel process). Tanaka's lemma on existence of the local time. (photos)
Lecture 26, Nov 24: Tanaka solution (with minimal filtration) does not admit a strong solutions started at zero. The concept of a weak solution. Lack of pathwise uniqueness for Tanaka equation. Abstract non-sense theory of Yamada and Watanabe: Uniqueness in holds under pathwise uniqueness and existence of a weak solution along with pathwise uniqueness imply existence of a solution map. Reduction of SDE with trivial martinagle term to an ODE. (photos)
Lecture 27, Nov 26: Methods for solving SDEs: Coordinate change (under time-homogeneity) to a "pure" SDE dX_t = σ(X_t)dB_t. Solving the "pure" SDE by time change. The Engelbert-Schmidt theorem. (photos)
Lecture 28, Dec 1: Exponential change of measure as a tool to change the mean. Exponential change for correlated Gaussians. The statement of Girsanov's theorem. The role of the condition EM_t=1 as a tool to ensure that M is a martingale. Proof of Girsanov's theorem. Application to a weak solution of SDE dX_t=a(t,X_t)dt+dB_t under minimal regularity conditions on t,x ↦ a(t,x). (photos)
Lecture 29, Dec 3: The importance, and subtlety, of the condition EM_t=1 in Girsanov's theorem. The condition fails for a function of a Bessel process. A sufficient condition via Hölder's inequality. The statement of Novikov's condition (which is just a cleaner sufficient and nearly necessary condition). Applications: solving SDEs and a proof that a standard Brownian motion conditioned to be positive is a 3-dimensional Bessel process. (photos)
Lecture 30, Dec 5: Occupation time measure for a general process. For one-dimensional Brownian motion, theTanaka formula suggest a density with respect to the Lebesgue measure. The Trotter existence theorem and its proof. Application to reflected Brownian motion and generalized Itô formula. (photos)