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### Syllabus for the Preliminary Examination in Probability

#### 1. Theory of Probability I - M385C

• Prerequisites:
• Real Analysis (M365C or equivalent),
• Linear Algebra (M341 or equivalent),
• Probability (M362K or equivalent).
• R. Durrett, Probability: theory and examples, third ed., Duxbury Press, Belmont, CA, 1996. (required)
• D. Williams, Probability with martingales, Cambridge University Press, Cambridge, 1991. (recommended)
• Literature:
• Syllabus:
(Note: all references are to Durrett's book)

Foundations of Probability:
• Random variables (Sections 1.1, 1.2): probability spaces, σ-algebras, measurability, continuity of probabilities, product spaces, random variables, distribution functions, Lebesgue-Stieltjes measures (without proof), random vectors, generation, a.s.-convergence
• Expected value (Section 1.3): abstract Lebesgue integration (without proofs), inequalities (Jensen, Cauchy-Schwarz, Chebyshev, Markov, Hölder, Minkowski), limit theorems (Fatou's lemma, monotone convergence and dominated convergence theorems), change-of-variables formula,
• Dependence (Section 1.4): independence, pairwise independence, Dynkin's - theorem, convolution of measures, Fubini's theorem, Kolmogorov's extension theorem (without proof)
Classical Theorems:
• Weak laws of large numbers (Sections 1.5, 1.6): the L2 -weak law of large numbers, triangular arrays, Borel-Cantelli lemmas, modes of convergence, inequalities (Markov, Chebyshev, Jensen, Hölder), the weak law of large numbers
• Central limit theorems (Sections 2.2, 2.3a, 2.3b, 2.3c, 2.4a, 2.9part ): weak convergence of distributions, the continuous mapping theorem, Helly's selection theorem, tightness, characteristic functions, the inversion theorem, continuity theorem, the central limit theorem, multivariate normal distributions
Discrete-Time Martingale Theory:
• Conditional expectation (Sections 4.1a, 4.1b): Radon-Nikodym theorem (without proof), conditional expectation, filtrations, predictability and adaptivity
• Martingales (Sections 4.2, 4.4, 4.5, 4.6part , 4.7): martingale transforms, the optional sampling the- orem, the upcrossing inequality, Doob's decomposition, Doob's inequality, Lp -convergence, maxi- mum inequalities, L2 -theory, uniform integrability, backwards martingales and the strong law of large numbers.

#### 2. Theory of Probability II - M385D

• Prerequisites:
• Graduate-level probability (M385C or equivalent).
• Literature:
• I. Karatzas and S. Shreve, Brownian motion and stochastic processes, second ed., Springer, 1991 (required)
• D. Revuz and M. Yor, Continuous martingales and stochastic processes, third ed., Springer, 1999 (recommended)
•
• Syllabus:
(Note: all references are to the book of Karatzas and Shreve)

Continuous-Time Martingale Theory:
• General theory of processes (Sections 1.1, 1.2) : Continuous-time processes and filtrations, types of measurability (optional, predictable, progressive), continuous stopping/optional times
• Path regularity of martingales (Section 1.3 A): existence of RCLL modifications, usual conditions for filtrations
• Convergence and optional sampling (Section 1.3 A-C): martingale inequalities, convergence theorems, optional sampling, uniform integrability and martingale with a last element
• Quadratic variation (Section 1.5 or Section IV.1 in Revuz-Yor): quadratic variation for continuous martingales, local martingales and localization, spaces of martingales
• Doob-Meyer decomposition (Section 1.4): no proof
Brownian Motion:
• Definition, construction and basic properties (Sections 2.1, 2.2): construction via Kolomogorov extension theorem, Hölder regularity of paths (Kolmogorov-Centsov), Gaussian processes
• The canonical space (Section 2.4): weak convergence on C[0, infinity), invariance principle, Wiener measure
• Markov and strong Markov property of Brownian motion (Sections 2.5-2.8, selected topics): reflexion principle, density of hitting times, Brownian filtrations, Blumenthal zero-one law
Stochastic Integration:
• Construction of the Stochastic Integral (Sections 3.1, 3.2): stochastic integration with respect to continuous local martingales, quadratic variation and Itô isometry
• Itô formula (Section 3.3): Itô formula, exponential martingales, linear stochastic differential equations
Applications (and extensions) of Itô's formula:
• Paul Léavy's characterization of Brownian motion (Section 3.3 B):
• Changes of measure (Section 3.5): Girsanov theorem, Brownian motion with drift Representations of martingales (Section 3.4): predictable representation property and Kunita-Watanabe decomposition, time-changed Brownian motions (Dambis-Dubins-Schwarz), Knight's theorem on orthogonal martingales
• Local time (Sections 3.6, 3.7): local time for Brownian motion and continuous semimartingales, Tanaka's formula, generalized Itô's formula for convex functions.