Let Y (t) be a real-valued process with stationary independent increments. Assume that the moment generating function ψ(α) = E[eαY (1)] exists for α in a neighborhood of 0, and g(t) = E[eαY (t)] is...

Let Y (t) be a real-valued process with stationary independent increments. Assume that the moment generating function ψ(α) = E[eαY (1)] exists for α in a neighborhood of 0, and g(t) = E[eαY (t)] is continuous at t = 0 for each α. Show that g(t) is continuous in t for each α, and that 396 5 Brownian Motion g(t) = ψ(α)t. Use the fact that g(t + u) = g(t)g(u), t, u ≥ 0, and that the only continuous solution to this equation has the form g(t) = etc, for some c that depends on α (because g(t) depends on α). Show that