### Syllabus for the Preliminary Examination in Probability

**(The first part of the Prelim exam will deal with the material covered in M385C and the second part of the Prelim exam will deal with the material covered in M385D)**

#### 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 L
^{2}-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, L
^{2}-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.