An elementary approach to Stochastic Differential Equations using the infinitesimals

TL;DR

This paper demonstrates how nonstandard analysis with infinitesimals offers a simple, elementary approach to stochastic differential equations, exemplified through deriving the Fokker-Planck equation for Brownian motion.

Contribution

It shows that elementary nonstandard calculus techniques suffice for stochastic differential equations, emphasizing the modeling advantages of infinitesimals over traditional methods.

Findings

Derivation of Fokker-Planck equation from naive Brownian motion model

Use of hyperfinite theory simplifies stochastic calculus

Nonstandard analysis provides a straightforward framework for SDEs

Abstract

The aim of this paper is to evidence two points relative to Nonstandard analysis (NSA): 1. In most applications of NSA to analysis, only elementary facts and techniques of nonstandard calculus seems to be necessary. 2. The advantages of a theory which includes infinitesimals rely more on the possibility of making new models rather than in the proving techniques. These two points will be illustrated in the theory of Brownian motion which can be considered as a classical model to test the power of the infinitesimal approach. Starting from a naive idea of Brownian motion, we deduce the Fokker-Plank equation in a simple and rigorous way. It is possible to keep every things to a simple level since all the theory of stochastic differential equations is treated as a hyperfinite theory and it is not translated in a "standard model". The only standard object is the final one: the Fokker-Plank equation.