Exercise 320 let snnbsp s0nbsp sigma fromnbspn to i1 nbspof

Question

Let B(t) be a Brownian motion. Show that the following processes are Brownian motions on [0, T ].

1. X(t) = -B(t).

2. X(t) = B(T - t) - B(T), where T < +∞.

3. X(t) = cB(t/c²), where T ≤ +∞.

4. X(t) = tB(1/t), t > 0, and X(0) = 0.

Exercise 3.20: Let S_n = S₀ + (sigma from n to i=1  of ξi) be a Random Walk, with P(ξ1 = 1) = p, P(ξ1 = -1) = 1 - p. Show that for any λ, e^γSn-λn is a martingale for the appropriate value of γ.