### Stochastic Processes (MATH136/STAT219, Winter 2019)

This course prepares students to a rigorous study of Stochastic Differential Equations, as done in Math236. Towards this goal, we cover -- at a very fast pace -- elements from the material of the (Ph.D. level) Stat310/Math230 sequence, emphasizing the applications to stochastic processes, instead of detailing proofs of theorems. A critical component of Math136/Stat219 is the use of measure theory.

The Stat217-218 sequence is an extension of undergraduate probability (e.g. Stat116), which covers many of the same ideas and concepts as Math136/Stat219 but from a different perspective (specifically, without measure theory). Thus, it is possible, and in fact recommended to take both Stat217-218 and Math136/Stat219 for credit. However, be aware that Stat217-218 alone is NOT adequate preparation for Math236.

Main topics of Math136/Stat219 include: introduction to measurable, Lp and Hilbert spaces, random variables, expectation, conditional expectation, uniform integrability, modes of convergence, stationarity and sample path continuity of stochastic processes, examples such as Markov chains, Branching, Gaussian and Poisson Processes, Martingales and basic properties of Brownian motion.

Prerequisites: Students should be comfortable with probability at the level of Stat116/Math151 (summary of material) and with real analysis at the level of Math115 (syllabus). Past exposure to stochastic processes is highly recommended.

Text: Download the course lecture notes and read each section of the notes prior to corresponding lecture (see schedule). When doing so, you may skip items excluded from the material for exams (see below) or marked as ``omit at first reading'' and all ``proofs''. Kevin Ross short notes on continuity of processes, the martingale property, and Markov processes may help you in mastering these topics.

Supplementary material: (texts on reserve at science library)

• Rosenthal, A first look at rigorous probability theory (accessible yet rigorous, with complete proofs, but restricted to discrete time stochastic processes).
• Grimmett and Stirzaker, Probability and Random Processes (with most of our material, in a friendly proof oriented style).
• Shreve, Stochastic Calculus for Finance II: Continuous time models, Ch. 1,2,3,A,B (covering same material as the course, but more closely oriented towards stochastic calculus).
• Karlin and Taylor, A first course in Stochastic Processes, Ch. 6,7,8 (gives many examples and applications of Martingales, Brownian Motion and Branching Processes).
• Lawler, Stochastic Processes (more modern examples and applications than in Karlin and Taylor).

Meeting: 200-305, Mo/We 1:30-2:50, replacement lectures: 380-380F, Fr 1:30-2:50, 1/25, 2/22 and 3/8.

Instructor: Amir Dembo, office hours: Seqouia 129, Mo 2:55-3:45 p.m. or e-mail adembo at stanford.edu (please include MATH136 in your email title).

CA: Panagiotis Lolas, office hours: 380-380T, Tu 3:00-5:00 p.m., We 10:00-11:00 a.m. or e-mail panagd at stanford.edu (please include MATH136 in your email title).

Grading : Judgement based on Final (50%) and Midterm (25%) exam marks and on consistent Homework efforts (25%). At least 60% required for CR grade.

Midterm: Tuesday 2/12, 6:00-7:30 p.m., Room 380-380C. Three pages of notes (2 sides each) allowed, handwritten or computer generated, at any font readable without artificial magnification. Material: Sections 1.1-3.3 of lecture notes, except: all of Section 2.2; from Section 2.4: up to 2.4.3; from Section 3.1: the cylindrical sigma-field; from Section 3.3: Fubini's theorem. practice exam and its solution.

Final: Wednesday, 3/20, 3:30-6:30, Room 380-380X. Six pages of notes (2 sides each) allowed, handwritten or computer generated, at any font readable without artificial magnification.

Material: Everything in lecture notes, except: all of Section 2.2; from Section 2.4: up to 2.4.3; Section 4.1.2; all of Section 6.3; everything marked as ``omit at first reading'' and all ``proofs'' unless done during lectures (80% of exam shall be from Sections 4.1--6.2).

Study tools: List of key items and Final-2019: Exam Format. Exercises 4.3.20, 4.4.6, 4.5.4, 4.6.7, 5.1.8, 5.2.6, 5.3.9, 6.1.19 and 6.2.12 are from previous finals. See also practice exam and its solution, or Final-2017: with solution.

Homework of 2019: Problems from the text as listed on HW1--HW9, are due each Wednesday 1:30 p.m. on a weekly basis ( Solutions: see Canvas page) Collaboration allowed in solving the problems, but you are to provide your own independently written solution. Please deliver your assignment to class on due date (late homework solutions will not be graded).

Schedule (Read corresponding sections of notes before class):

```1/7         Mo(1.1/1.2.1)             We(---)
1/14        Mo(1.2.1/1.2.2/1.2.3)     We(1.3.1/1.4.1)
1/21        Mo(---)                   We(1.3.2/1.4.2)     Fr(1.4.3/2.1/2.3)
1/28        Mo(2.3/2.4)               We(3.1)
2/4         Mo(3.2.1/3.2.2)           We(3.2.3/3.3/5.1)
2/11        Mo(Review:1-3)            We(5.1/4.1.1/4.1.3)
2/18        Mo(---)                   We(4.1.3/4.2)       Fr(4.3.1/4.3.2)
2/25        Mo(5.2/4.4.1)             We(4.4.2/4.5/4.6)
3/4         Mo(4.6/5.3)               We(6.1)             Fr(6.1/6.2)
3/11        Mo(6.2)                   We(Review:4-6)
```

Approximately equivalent material (outdated):

• Grimmett and Stirzaker: 1.1-1.3, 1.5-1.6, 2.1, 2.3-2.5, 3.1-3.3, 3.5, 3.7, 4.1-4.6 (partially), 5.4, 5.6-5.9, most of 7.1-7.3 (10), 8.1, 8.2, 8.5, 8.6, 9.1 (=midterm), 9.6, 7.7-7.10, 12.1, 12.3-12.8, 13.4 (without most proofs).
• Shreve's book: 1.1-1.5, 2.1-2.6.

List of key items:

• Probability spaces, generated and Borel sigma-algebras. Indicators, simple functions, random variables. Expectation: Lebesgue and Riemann integrals, motonicity and linearity. Jensen's and Markov's inequalities. L_q spaces. Independence. Distribution, density and characteristic function. Convergence almost surely, in probability, in q-mean and in distribution/law (=weakly). Uniform integrability, Dominated and Monotone convergence. Conditional expectation: definition and properties.
• Stochastic processes: definition, stationarity, finite-dimensional distributions, version and modification, sample path continuity, right-continuous with left-limits processes. Kolmogorov's continuity theorem and Holder continuity. Stopping times, stopped sigma-fields and processes. Right-continuous and canonical filtrations, adapted and previsible processes.
• Examples: random walk; Gaussian distribution: for variables, vectors and processes, non-degeneracy, stationarity, closeness under 2-mean convergence.
• Brownian motion: definition, Gaussian construction, independence of increments, scaling and time inversion, Levy's martingale characterization, reflection principle, law of its maximum in an interval and first hitting time of positive levels, modulus of continuity, quadratic and total variation. Related processes: Geometric Brownian motion, Brownian bridge and Ornstein-Uhlenbeck process.
• Poisson distribution, approximation, and process: definition, rate, construction, independence of increments, memoryless property of the Exponential law, the dual process of independent Exponential interarrivals, the order statistics of independent uniform samples.
• Markov chain and process: Markov and strong Markov property, examples.
• Discrete and continous time martingales: definition, superMG and subMG, convex functions of, stopped MG and the martingale transform, existence of RCLL modification, Doob's optional stopping, representation, inequalities and convergence theorems, examples - Doob's martingale and martingales derived from random walk, Brownian motion, branching and Poisson processes.