Introduction to Stochastic Analysis: Integrals and Differential Equations

Chapter 8

Stratonovich Integral and Equations

 

 

Non-differentiability of the trajectories of Brownian motion, stochastic integration theory, and accompanying “exotic” formulas of Itô, integration by parts, etc. is the price we pay for the mathematical idealization of various real-world models with perturbations close to white noise. However, our troubles do not end here. Suppose that, in some real experiment, we observe the integral with “true” Brownian motion, and wish to construct its realistic mathematical model. Although the trajectories of a “true” Brownian motion, as those of its mathematical model, are rather chaotic, they are necessarily differentiable.¹ To get an idealized model of the integral we may try to take a sequence of continuously differentiable functions Bⁿ, , converging to a Brownian motion B and consider the corresponding sequence of integrals . Because of the differentiability of Bⁿ, these integrals are . Since lim in the real world, the above-mentioned integral ...