Question: a let mt be a martingale under a probability measure...
(a) Let (Mt) be a martingale under a probability measure Q and let g be a deterministic function. Assume that g is not constant in t. Is the process
Ut = Mt + g(t) a martingale under Q? Justify your argument.
(b) Let βt = ert be the value of the savings account at time t. Consider the model for the stock price
St = Xt + 0.1esin(t) where (Xt) solves the Black-Scholes SDE
dXt = rXtdt + σXtd(Bt_hat), X0 = 1
Here (Bt_hat) is a Brownian motion under Q which turns (Xt/βt) into a martingale. Show that (St/βt) is NOT a martingale under Q.
Hint: The result of part a will help!