ABSTRACT
In applied science white noise is often taken as an idealization of phenomena involving sudden and extremely large fluctuations. Mathematically, one can think of white noise as a stochastic process z(i) such that zffls are independent and for each ¿, z(t) has mean 0 and variance oo in the sense that
(3.1.1) where 6 is the Dirac delta function. Thus it seems to be reasonable to claim that we can define an integral J f(t)z(t) dt such that
But what is the definition of the integral
3.1.2 White noise as the derivative of a Brownian motion White noise can be regarded as the derivative of a Brownian motion. But what is a Brownian motion? As is well-known, Robert Brown made microscopic observations in 1827 that small particles contained in the pollen of plants, when immersed in a liquid, exhibit highly irregular motions. This highly irregular motion is called a Brownian motion. Mathematically, a
But then this means that the derivative of B(t), or the white noise B(t), does not exist. Hence the integral fa f(t)B(t) dt does not seem to be defined at all.
