14.16 [Doobs inequalities] For a stochastic process X = {Xn}n with a filtration = {n}n, we define the maximum process X n() = max

14.16 [Doob’s inequalities] For a stochastic process X = {Xn}n with a filtration

ℱ = {ℱn}n, we define the maximum process X¯ n(ω) = max 0≤i≤n Xi(ω).

a) Suppose the original process X = {Xn}n is a submartingale. Show that P(X¯ n ≥ λ) ≤

1

λE )

Xn1X¯n≥λ

1

≤

1

λE )

Xn1X¯n≥0 1

= E )

X+

n 1