Show that {Y (t), t 0} is a Martingale when Y (t) = exp{cB(t) − c2t/2}

where c is an arbitrary constant. What is E[Y (t)]?

An important property of a Martingale is that if you continually observe the process and then stop at some time T , then, subject to some technical conditions

(which will hold in the problems to be considered), E[Y (T )] = E[Y (0)]

The time T usually depends on the values of the process and is known as a stopping time for theMartingale. This result, that the expected value of the stopped Martingale is equal to its fixed time expectation, is known as the Martingale stopping theorem.