Date of Defense
22-4-2025 11:00 AM
Location
F3-022
Document Type
Dissertation Defense
Degree Name
Doctor of Philosophy in Mathematics
College
COS
Department
Mathematical Sciences
First Advisor
Prof. Farrukh Mukhamedov
Keywords
Quadratic Stochastic Operator, Quadratic Stochastic Process, associativity, character, dynamics, Markov process, ergodicity coefficient, Susceptible Infected Recovered model, automorphisms, derivations, Rota Baxter operator, option pricing, Equivalent Martingale Measure, infinitesimal generator.
Abstract
This research focuses on the algebraic structures of the Quadratic Stochastic Processes (QSPs). In this work, we first study π(π)- Quadratic Stochastic Operators (QSOs) linked to the partition β3. We simultaneously discuss the dynamics of the obtained QSOs. Moreover, the algebraic structure of the associated genetic algebra is studied. Further, we build Quadratic Stochastic Processes (QSPs) using the given Markov processes. Consequently, we obtain an ordinary differential equation for the resultant Quadratic Stochastic Processes (QSPs). Besides, we apply the solution of this ordinary differential equation to the option pricing problem. Thereafter, we construct Quadratic Stochastic Processes (QSPs) in three-dimensional space by utilizing the parameters of the Susceptible-Infected-Recovered (SIR) model. In addition, we investigate the algebraic properties of the limiting genetic algebras. Rota-Baxter operators are also analyzed for different weights for these algebras. In the application part of this analysis, we propose an option pricing under the Quadratic Stochastic Process (QSP) modulated Geometric Brownian Motion (GBM) model. We also analyse the Radon-Nikodym derivative of the Equivalent Martingale Measure (EMM) π with respect to the historic probability π. Ultimately, we obtain an infinitesimal generator which facilitates numerical simulations of the non-Markovian and stock price processes.
QUADRATIC STOCHASTIC PROCESSES: ALGEBRAIC STRUCTURES AND THEIR APPLICATIONS
F3-022
This research focuses on the algebraic structures of the Quadratic Stochastic Processes (QSPs). In this work, we first study π(π)- Quadratic Stochastic Operators (QSOs) linked to the partition β3. We simultaneously discuss the dynamics of the obtained QSOs. Moreover, the algebraic structure of the associated genetic algebra is studied. Further, we build Quadratic Stochastic Processes (QSPs) using the given Markov processes. Consequently, we obtain an ordinary differential equation for the resultant Quadratic Stochastic Processes (QSPs). Besides, we apply the solution of this ordinary differential equation to the option pricing problem. Thereafter, we construct Quadratic Stochastic Processes (QSPs) in three-dimensional space by utilizing the parameters of the Susceptible-Infected-Recovered (SIR) model. In addition, we investigate the algebraic properties of the limiting genetic algebras. Rota-Baxter operators are also analyzed for different weights for these algebras. In the application part of this analysis, we propose an option pricing under the Quadratic Stochastic Process (QSP) modulated Geometric Brownian Motion (GBM) model. We also analyse the Radon-Nikodym derivative of the Equivalent Martingale Measure (EMM) π with respect to the historic probability π. Ultimately, we obtain an infinitesimal generator which facilitates numerical simulations of the non-Markovian and stock price processes.
