[Doob’s Stopping Time Theorem: you cannot make money from a martingale by random stopping] (i) Let (X n , F n ) n=1,2,.,N be a martingale in discrete…

[Doob’s Stopping Time Theorem: you cannot make money from a martingale by random stopping]

(i) Let (Xn, Fn)n=1,2,…,N be a martingale in discrete time on the integers {1, 2, . . . , N} for a finite N. Show that E(Xτ ) = E(X1). [Hint: the events τ = n form a partition of the sample space.]

(ii) Next formulate and prove (extreme rigour not required) a version in continuous time.