ABSTRACT
This chapter introduces the stochastic integral, stochastic differential equations, and core results of Ito calculus.
4.1 Wiener process (Brownian motion)
Let T>0 be given, Definition 4.1 We say that a continuous time random process w(t) is a (one-
dimensional) Wiener process (or Brownian motion) if
(i) w(0)=0; (ii) w(t) is Gaussian with Ew(t)=0, Ew(t)2=t, i.e., w(t) is distributed as N(0, t); (iii) w(t+τ)−w(t) does not depend on {w(s), s≤t} for all t≥0, τ>0.
