Let W(t) be the Wiener process. Then the increment W(t) −W(s) over the time interval (s,t] is normally distributed with mean zero and variance t − s, and the increments over disjoint time intervals...

Let W(t) be the Wiener process. Then the increment W(t) −W(s) over the time interval (s,t] is normally distributed with mean zero and variance t − s, and the increments over disjoint time intervals are independent. Let V(t) be a strictly increasing continuous function with V(0) = 0, and introduce the stochastic process U(t) = W(V(t)). Finally let
 be generated by U(s) for

a) Prove that
  for all
  that is, that U(t) is a martingale.

b) Prove that
  that is, that U²
(t) −V(t) is a martingale. Note that this shows that

Chapter 3