Brownian motion is a stochastic process of great theoretical importance, and as the basic building block of a variety of other processes, of great practical importance as well. In this chapter we stud...Brownian motion is a stochastic process of great theoretical importance, and as the basic building block of a variety of other processes, of great practical importance as well. In this chapter we study Brownian motion and a number of random processes that can be constructed from Brownian motion. We also study the Ito stochastic integral and the resulting calculus, as well as two remarkable representation theorems involving stochastic integrals.
This is a statement of time homogeneity: the underlying dynamics (namely the jostling of the particle by the molecules of water) do not change over time, so the distribution of the displacement of the...This is a statement of time homogeneity: the underlying dynamics (namely the jostling of the particle by the molecules of water) do not change over time, so the distribution of the displacement of the particle in a time interval \( [s, t] \) depends only on the length of the time interval.
