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

15Synthesis

We conclude this book with a brief synopsis of the main points it has covered, using some informality that skips the more technical issues.

Stochastic differential equations (SDEs) are useful tools in several scientific and technological fields in order to model dynamical phenomena that would usually be described by ordinary differential equations if it were not for the fact that their dynamics are affected by a perturbing noise. The perturbing noise, a stochastic process in continuous time, can, for the sake of mathematical convenience, be approximated by a white noise , which is a generalized stationary Gaussian stochastic process whose values at different times are independent. It is the derivative (in the sense of generalized functions since the ordinary derivative does not exist) of the Wiener process (also known as Brownian motion) (), usually abbreviated to . Reversely, we may say that is the integral of the white noise, and so it represents the accumulated noise up ...