QUADRATIC STOCHASTIC PROCESSES: ALGEBRAIC STRUCTURES AND THEIR APPLICATIONS

Abstract

This research focuses on the algebraic structures of the Quadratic Stochastic Processes (QSPs). In this work, we first study ()- Quadratic Stochastic Operators (QSOs) linked to the partition ℙ3. 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) with respect to the historic probability . Ultimately, we obtain an infinitesimal generator which facilitates numerical simulations of the non-Markovian and stock price processes