# Stochastics

# Multiplicative LLN and CLT and their Applications

## 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.

# Stochastic Stability of Planar Flows

## Publication: “*Stabilization by noise of a C^2-valued coupled system.*” Joe P. Chen, Lance Ford, Derek Kielty, Rajeshwari Majumdar, Heather McCain, Dylan O’Connell, and Fan Ny Shum. Stochastics and Dynamics Vol. 17, No. 06, 1750046 (2017)

## Project

# Stochastic Stability of Planar Flows

## Publication: arXiv:1510.09221

Journal reference: | Stochastics and Dynamics, Vol. 17, No. 6 (2017) 1750046 |

DOI: | 10.1142/S0219493717500460 |

## Event

## Presentation

## Project

# Stochastic Stability of Planar Flows

## Publication: arXiv:1510.09221

Journal reference: |
Stochastics and Dynamics, Vol. 17, No. 6 (2017) 1750046 |

DOI: |
10.1142/S0219493717500460 |

## Group Members

Lance Ford, Derek Kielty, Rajeshwari Majumdar, Heather McCain, Dylan O’Connell

## Supervisors

## Overview

We investigated systems of complex-valued ordinary differential equations (ODEs) that blows up in finite time, which we refer to as *explosive* systems. The goal is to understand for what initial conditions does the system explode and will the addition of noise stabilize it; that is, if we were to perturb the system with an additive Brownian motion, will the system of stochastic differential equation (SDE) still be explosive? In fact, we were able to prove a toy model of the stochastic Burgers’ equation to be ergodic; that is, the SDE is nonexplosive and it has a unique limiting distribution.

## Presentation

# Lyapunov Exponents of Multiplicative Stochastic Processes

## Group Members

David Wierschen and Becky Simonsen

## Overview

This group considered the stability of matrix Lie group valued stochastic differential equations, dX_{t} = AX_{t}dt + BX_{t}dt. Random dynamical systems such as this arise in many applications (e.g., oceanic turbulence, helicopter blade motion, light in random channels, wireless networks) in which stability is of practical and theoretical concern. The stability of the zero solution, X_{t} = 0, is determined by the top Lyapunov exponent. But in practice, analytic calculations of the Lyapunov exponent are often impossible, so time discrete approximations and simulations are necessary. Oceledet’s famous multiplicative ergodic theorem ensures that the Lyapunov exponent of X_{t} is almost surely constant. But the Lyapunov exponent of a time discrete approximation is itself a random variable. The mean of this random variable has been studied. The group provided estimates on the variance, distribution and rate of convergence in certain numerical approximation methods. In addition, they expanded on and provided simulations for recent results regarding the top Lyapunov exponent of certain Lie group valued SDEs.

# The Top Lyapunov Exponent of Sp(2n,R)-Valued Multiplicative Stochastic Processes

# 3rd Northeast Mathematics Undergraduate Research Mini-Symposium

**Organizers**

#### Participating Schools: Amherst, Colgate, Smith, UConn, UMass, Williams

#### 3rd Mini-Symposium full program (2015)

(photo courtesy Megan Brunner)

Mini-Symposium poster