This exercise deals with obtaining martingales. Suppose X_t is a geometric process with drift ? and diffusion parameter ?.

(a) When would the e^-rt X_t be a martingale? That is, when would the following equality hold.

(b) More precisely, remember from the previous derivation that

Or, again,

Which selection of ? would make e^?rt?X_t a martingale? Would

? = r

work?

(c) How about

? = r + ?²?

(d) now try:

? = r ? ? ?².

Note that each one of these selections defines a different distribution for the e^?rt X_t.

E" [e-"X,\X,s < ] =e"X,- E[e*X, X. s < t] - X,e"elut!e"xt-) = X,e-"e "t-lu+!ae-) E[e "X,\X,s