The Standard Brownian Motion on \([0, \infty )\) was already introduced in the last chapter. Since one sometimes talks about Standard Brownian Motion on a closed bounded interval [0, T] also, we start by redefining SBM, for the sake of records. In what follows I will denote either \([0, \infty )\) or the closed bounded interval [0, T] for some \(T\in (0, \infty )\) . Also, a family \(\{X_t, t\in I\}\) of real random variables on a probability space will be called a real “stochastic process”, indexed by I. It often helps to think of the index set I as representing “time” and the stochastic process \(\{X_t, \, t\in I\}\) as modelling the random motion of a particle in one-dimension with time. Also, following the standard and commonly used notation. We will usually denote a SBM by \(\{B_t,\, t\in I\}\) .