Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance

1Introduction

Stochastic differential equations (SDEs) are basically differential equations with an additional stochastic term. The deterministic term, which is common to ordinary differential equations, describes the ‘average’ dynamical behaviour of the phenomenon under study and the stochastic term describes the ‘noise’, i.e. the random perturbations that influence the phenomenon. Of course, in the particular case where such random perturbations are absent (deterministic case), the SDE becomes an ordinary differential equation.

As the dynamical behaviour of many natural phenomena can be described by differential equations, SDEs have important applications in basically all fields of science and technology whenever we need to consider random perturbations in the environmental conditions (environment taken here in a very broad sense) that affect such phenomena in a relevant manner.

As far as I know, the first SDE appeared in the literature in Uhlenbeck and Ornstein (1930). It is the Ornstein–Uhlenbeck model of Brownian motion, the solution of which is known as the Ornestein–Uhlenbeck process. Brownian motion is the irregular movement of particles suspended in a fluid, which was named after the botanist Brown, who first observed it at the microscope in the 19th century. The Ornstein–Ulhenbeck model improves Einstein treatment of Brownian motion. Einstein (1905) explained the phenomenon by the collisions of the particle with the molecules of the fluid and provided a model for the ...