### Stochastic Processes (MATH136/STAT219, Winter 2021)

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 can not replace Math136/Stat219 as preparation for a study of Stochastic Differential Equations (i.e. 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. 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''. Alternatively, view prior to each lecture the relevant pre-recorded annotated reading from the notes, or go over the slides for each lecture (as posted on Canvas). Kevin Ross short notes on continuity of processes, the martingale property, and Markov processes may help you in mastering these topics.

Supplementary material: (online, or 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: Tu/Th 8:30-9:50pm (except: Tu 3/16 5:30-7:00pm). Synchronous recorded discussions, with breakout, anonymous polling and a TA monitored chat-line, serving also as instructor's (public) office hours.

Instructor: Amir Dembo. For questions on material, use our Piazza page, or TA office hours, or e-mail adembo at stanford.edu (with MATH136/STAT219 as subject), for setting a (confidential) private office-hours meeting.

CA1 (HW1/HW3/HW5/HW7/HW9; chat Tue meetings): Sky Cao, office hours Mo 12:00-1:30pm, Tu 12:00-1:30pm (to 3/17) or e-mail skycao at stanford.edu (with MATH136/STAT219 as your email subject).

CA2 (HW2/HW4/HW6/HW8/HW9; chat Thu meetings): Youngtak Sohn, office hours Fr 5:30-7:00pm, Mo 5:30-7:00pm, (to 3/16) or e-mail youngtak at stanford.edu (with MATH136/STAT219 as your email subject).

Grading : Judgement based on two Midterm exam marks (36% each) and on consistent Homework (22%) and Participation (6%) efforts (see Panopto recorded introduction on Canvas). At least 60% required for CR grade.

Midterm 1: Open books, timed 1.5h exam, taken via Gradescope within a 16h frame starting 6:00am PST on Th 2/18 (upload frame ends 10:00pm PST, Th 2/18).

Material: Sections 1.1-3.3 and 5.1 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+solution posted on Canvas).

Midterm 2: Open books, timed 1.5h exam, taken via Gradescope within a 16 frame starting 6:00am PST on Th 3/18 (upload frame ends 10:00pm PST, Th 3/18).

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 Sections 6.2-6.3; everything marked as ``omit at first reading'' and all ``proofs'' unless done during lectures (over 90% of exam shall be from Sections 4.1--6.2). Practice Exam Posted (Canvas), solutions provided 3/16 (via Gradescope).

Study tools: List of key items, Exercises 4.3.20, 4.4.6, 4.5.4, 4.6.7, 5.1.8, 5.2.6, 5.3.9 and 6.1.19 are from previous Midterm2 exams.

Homework of 2021: Problems from the text as listed on HW1--HW9 ( Posted! ), are to be submitted through Gradescope each Tuesday at 6:30pm (no grading of late submissions). Collaboration allowed in solving the problems, but you are to provide your own independently written solution. Your assignment will typically be graded and returned on Gradescope the following week. Solutions are posted (on the course Canvas page), within 48h of due date.

Schedule (Read corresponding sections of notes before class):

```1/11       Tu(1.1/1.2.1/1.2.2)       Th(1.2.3/1.3.1)
1/18       Tu(1.3.2/1.4.1)           Th(1.4.2/1.4.3/2.1)
1/25       Tu(2.1/2.3)               Th(2.4/3.1)
2/1        Tu(3.2.1/3.2.2)           Th(3.2.3/3.3)
2/8        Tu(5.1/4.1.1)             Th(4.1.1/4.1.3)
2/15       Tu(Review:1-3+5.1)        Th(Asynchronous:4.2/4.3.1)
2/22       Tu(4.3.1/4.3.2)           Th(4.4.1/5.2)
3/1        Tu(5.3/4.4.2)             Th(4.5/4.6)
3/8        Tu(6.1)                   Th(6.1/6.2)
3/15       Tu(Review:4-6)            Th(--)
```

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, monotonicity 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.
• 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.
• Poisson distribution, approximation, and process: definition, rate, construction, independence of increments, memoryless property of the Exponential law, the dual process of independent Exponential inter-arrivals, the order statistics of independent uniform samples.