Stochastic Calculus
3
In person
Tuesdays and Thursdays 12:30-1:50 p.m.
Wachman 617
Wachman 612
Tuesdays 2–3 p.m., Thursdays 2–4 p.m.
L.-P. Arguin, A First Course in Stochastic Calculus, Pure Appl. Undergrad. Texts, 53, AMS, 2022.
Homework (weekly) 20%; numerical projects (weekly) 20%; midterm exam 24%; final exam 36%.
MATH 2043 (Calculus III, i.e., multivariable calculus), MATH 2101 (Linear Algebra), MATH 3031 (Probability Theory I), or equivalent. Prior knowledge of basic stochastic processes or advanced calculus (i.e., mathematical analysis) can be helpful but is not assumed.
To introduce the theory of Ito calculus and stochastic differential equations based on Brownian motion and Gaussian processes (avoiding concepts and results from measure theory whenever possible); to illustrate the theory with concrete examples and numerical projects; to present the applications of the theory to option pricing in finance.
Basic notions of probability; Gaussian processes; properties of Brownian motion; Martingales; Ito calculus; multivariate Ito calculus; Ito processes and stochastic differential equations; the Markov property; change of probability; applications to mathematical finance.
The date of the midterm exam will be announced at least two weeks in advance. The final exam will be during the week officially designated by the University. Both exams will be in class and closed book.
See below.
A problem set will be assigned at the end of each week (on Canvas), to be submitted in one week (in person). Late homework will not be accepted. At the end of the semester, the lowest homework score of each student will be dropped.
Small numerical projects will be assigned every week (on Canvas). Students will write the code in Python in a Jupyter notebook. The coding is elementary and the necessary Python knowledge can be picked up along the way.
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