Stochastics

Fractional Gaussian fields on surfaces and graphs

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 will 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).

 

Hedging by Sequential Regression in Generalized Discrete Models and the Follmer-Schweizer decomposition

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

The financial value of knowing the distribution of stock prices in discrete market models

The financial value of knowing the distribution of stock prices in discrete market models
Ayelet Amiran, Fabrice Baudoin, Skylyn Brock, Berend Coster, Ryan Craver, Ugonna Ezeaka, Phanuel Mariano and Mary Wishart
Vol. 12 (2019), No. 5, 883–899
DOI: 10.2140/involve.2019.12.883

arXiv:1808.03186

 

 

project page:

Financial Math: Portfolio Optimization and Dynamic Programming

Math UConn REU at JMM 2018

Two of our REU (2017 Stochastics) participants, Raji Majumdar and Anthony Sisti, will be presenting 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 both of them will be giving talks on Saturday, January 13 at the AMS Contributed Paper Session on Research in Applied Mathematics by Undergraduate and Post-Baccalaureate Students.

Their travel to the 2018 JMM has been made possible with the support of the MAA and UConn’s OUR travel grants.

Financial Math: Portfolio Optimization and Dynamic Programming

Group Members

Ayelet Amiran, Skylyn Brock, Ryan Craver, Ugonna Ezeaka, Mary Wishart 

Supervisors

Fabrice Baudoin, Berend Coster, Phanuel Mariano

 

Overview

Financial markets have asymmetry of information when it comes to the prices of assets. Some investors have more information about the future prices of assets at some terminal time. However, what is the value of this extra information?

We studied this anticipation in various models of markets in discrete time and found (with proof) the value of this information in general complete and incomplete markets. For special utility functions, which represent a person’s satisfaction, we calculated this information for both binomial (complete) and trinomial (incomplete) models.

Publication

Journal reference: Involve 12 (2019) 883-899
DOI: 10.2140/involve.2019.12.883

arXiv:1808.03186

Presentation

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Poster

PosterFinMath2018

Multiplicative LLN and CLT and their Applications

Group Members

Lowen PengAnthony SistiRajeshwari 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.