Discrete Calculus : Applications in Stochastic Processes

Abstract

In this dissertation we explore the relationship between the infinitesimal and discrete descriptions of Nature and how these descriptions are connected in a systematic way via an integral transform called mimetic map, which we propose. We start presenting a brief review of sequences and difference equations, detailing some solving methods. In particular, we see that solving techniques of differential equations of infinitesimal calculus can be transferred to a calculus used to describe and solve finite difference equations, known as discrete calculus, been such techniques widely applied in the research field of difference equations. Then we show how the whole structure of infinitesimal calculus can be transferred to the discrete calculus via the mimetic map, generalizing and systematizing the already known discrete calculus of sequences, using the discrete functions, and interpreting the difference equations as discrete versions of differential equations. Also via the mimetic map we extend the notion of generating functions of sequences to discrete functions, where such extensions depend on a parameter 𝑕, returning the sequence case when 𝑕 “ 1. With the mimetic map as well we obtain discrete versions of integral transforms, such as the discrete Laplace and Mellin transforms, relating the former with the Z transform. We also present a complex mimetic map used to construct a complex discrete calculus starting from the calculus on the complex plane. As applications in physics, we present a review of discrete and continuous stochastic processes and show how the mimetic transform and the corresponding discrete calculus are capable to map the descriptions of these processes into one another continuous processes one onto the other. In particular, we obtain a discrete version of the H theory for the background variables using the mimetic map and for the observable variable using the tools of stochastic processes. And lastly we present how the formulations of epidemic models, given as continuous and discrete stochastic processes, which is connected by construction in the literature, now could be connected via the discrete calculus and the mimetic map.