Group Members: Tyler Campos, Andrew Gannon, Benjamin Hanzsek-Brill, Connor Marrs, Alexander Neuschotz, Trent Rabe and Ethan Winters. 

 

Mentors: Rachel Bailey, Fabrice Baudoin, Masha Gordina

Overview: We study and simulate on computers the fractional Gaussian fields and their discretizations on surfaces like the two-dimensional sphere or two-dimensional torus. The study of the maxima of those processes will be done and conjectures formulated concerning limit laws. Particular attention will be paid to log-correlated fields (the so-called Gaussian free field).

“Computing Extreme Values of Continuous & Discrete Fractional Gaussian Fields on Manifolds” by Tyler Campos (Yale & UConn REU 2022) and Connor Marrs (Bowdoin & UConn REU 2022)
“Paths of the Fractional Gaussian Field on S1 and the Torus” by Andrew Gannon (University of Connecticut)

 

 

William Busching, Delphine Hintz, Oleksii Mostovyi, Alexey Pozdnyakov

Involve (2022), Vol. 15(4), pp. 649-668. [published version] [arXiv]

 

Jake Koerner, Joo Seung Lee, Oleksii Mostovyi

preprint

accepted in the Missouri Journal of Mathematical Sciences (2023)

Group Members

Sarah Boese, Tracy Cui, Sam Johnston

Supervisors

Gianmarco Molino, Olekisii Mostovyi

Overview

In practice, financial models are not exact — as in any field, modeling based on real data introduces some degree of error. However, we must consider the effect error has on the calculations and assumptions we make on the model.  In complete markets, optimal hedging strategies can be found for derivative securities; for example, the recursive hedging formula introduced in Steven Shreve’s “Stochastic Calculus for Finance I” gives an exact expression in the binomial asset model, and as a result the unique arbitrage-free price can be computed at any time for any derivative security.

In incomplete markets this cannot be accomplished; one possibility for computing optimal hedging strategies is the method of sequential regression.  We considered this in discrete-time; in the (complete) binomial model we showed that the strategy of sequential regression introduced by Follmer and Schweizer  is equivalent to Shreve’s recursive hedging formula, and in the (incomplete) trinomial model we both explicitly computed the optimal hedging strategy predicted by the Follmer-Schweizer decomposition and we showed that the strategy is stable under small perturbations.

Publication “Stability and asymptotic analysis of the Föllmer–Schweizer decomposition on a finite probability space” Involve, a Journal of Mathematics , v.13 , 2020 doi.org/10.2140/involve.2020.13.607

Presentation

Poster

Group Members

Lowen Peng, Anthony Sisti, Rajeshwari Majumdar

Supervisors

Phanuel Mariano, Masha Gordina, Sasha Teplyaev, Ambar Sengupta, Hugo Panzo

Overview

We study the Law of Large Numbers (LLN) and and Central Limit Theorems (CLT) for products of random matrices. The limit of the multiplicative LLN is called the Lyapunov exponent. We perturb the random matrices with a parameter and we look to find the dependence of the the Lyapunov exponent on this parameter. We also study the variance related to the multiplicative CLT. We prove and conjecture asymptotics of various parameter dependent plots.

Publication: “Lyapunov exponent and variance in the CLT for products of random matrices related to random
Fibonacci sequences” — arXiv:1809.02294, Discrete Contin. Dyn. Syst. Ser. B 25 (2020), pp 21

Presentations:

Raji Majumdar and Anthony Sisti, will present posters Applications of Multiplicative LLN and CLT for Random Matrices and Black Scholes using the Central Limit Theorem on Friday, January 12 at the MAA Student Poster Session, and give talks on Saturday, January 13 at the AMS Contributed Paper Session on Research in Applied Mathematics by Undergraduate and Post-Baccalaureate Students.