This is the webpage for the course MATH 136 – Stochastic Processes.

**IMPORTANT NEWS: **The first 3 weeks of lectures will be ONLINE on Zoom (but not recorded!). From Week 4 on, we will return to have in-presence lectures.

The office hours of the course assistant Panagiotis Lolas will be in-presence starting from Week 4 (but online on Week 1-2-3).

This plan might change according to possible modifications in Stanford mandates and travel restrictions.

**Lecture times:** Tuesday+Thursday 1:30 PM -3:00 PM (380-380F)

**Instructor:** Jacopo Borga, jborga_at_stanford.edu **Office hours:** Thursday 8:30-9:30 AM.

**Assistant:** Panagiotis Lolas, panagd_at_stanford.edu **Office hours:** Monday 4:00 PM -6:00 PM, Friday 4:00 PM -6:00 PM.

**Description:** 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.

**Program**: We will mainly follow this lecture notes written by Prof. Amir Dembo. There is no required textbook. Here is a tentative program for the course (the numbers below refer to the Sections in the lecture notes):

*Week 1:* Tu(1.1/1.2.1/1.2.2) Th(1.2.3)

*Week 2:* Tu(1.3) Th(1.4)

*Week 3:* Tu(2.1) Th(2.3/2.4)

*Week 4: *Tu(3.1/3.2.1/3.2.2) Th(3.2.3/3.3)

*Week 5:* Tu(5.1/4.1.1) Th(Midterm exam)

*Week 6:* Tu(4.1.1/4.1.3) Th(4.2/4.3.1)

*Week 7: *Tu(4.3.1/4.3.2) Th(4.4.1/5.2)

*Week 8:* Tu(5.3/4.4.2) Th(4.5/4.6)

*Week 9:* Tu(6.1) Th(6.1/6.2)

*Week 10:* Tu(Review:4-6) Th(–)

**Exercises: **A new homework (with problems/exercises from the lecture notes) will be posted at the start of every week (or earlier), and should be submitted on Gradescope by 10PM the following Tuesday. I will set up Gradescope to actually accept submissions until a day later, as a grace period in case of last-minute technical difficulties. Any other delay will be not accepted (i.e., no grading for late submissions). Also, your lowest homework score will be dropped from consideration.

You are permitted (and encouraged!) to discuss the problems with other students, but you must write up the solutions yourself. Please work out problems neatly — do not hand in your scratch work.

**Midterm exam: **Thursday, February 3 at 1:30 PM -3:00 PM.

**Final exam:** Tuesday, March 15 at 3:30 PM-6:30 PM.

**Grading: **Judgement based on Midterm exam mark (30%), Final exam mark (40%) and on consistent Homework (30%).Panagiotis Lolas