On applications of stochastic differential equations

Abstract

The study of dynamic systems represents a large field of mathematics that models the temporal evolution of phenomena in real life. The theory of differential equations largely deals with the modeling of these systems through the laws that govern the temporal rates at which the states describing dynamic systems change. In this context, we study those models in which a random factor is considered in the law that governs the rate of change of these systems over time. These are called stochastic differential equations. This work briefly presents the mathematical theory underlying this type of differential equation by constructing an analogy between the theory of ordinary differential equations and that of stochastic ones. Finally, and as the main objective, two applications of the theory are presented in problems at the state of the art: stochastic thermodynamics and probabilistic numerical methods. The first deals with an attempt to extend the classical theory of thermodynamics of equilibrium systems to systems out of equilibrium, such as chemical reactions presented in this work, for example. In the second problem, stochastic differential equations are used to probabilistically model numerical algorithms for solving ordinary differential equations, which can even account for uncertainties that the modeler might have regarding the system to be studied. These theories are reviewed and applied to systems already known in the literature. In the case of stochastic thermodynamics, we apply the theory to the study of enzymatic reactions governed by the Michaelis-Menten mechanism. For probabilistic numerical methods, we apply the theory to try to identify the parameters of a bioreactor model, which is part of a model of pilot wastewater treatment plant used as a test bench for water treatment systems.