Bonus question. (5+5+5+5 points) Let W be a standard Brownian motion on (, (F)t>0, F, P) and consider the Brownian motion with drift m e R.
X=W+mt
t > 0.
(a) Find X e R such that e^Xt is a martingale
(b) For a < 0 < b, use (a) to compute the probability that X reaches a before b
Ta,b:=inf{t>0:X=a or X=b}
is P-a.s. finite.
(c) Argue that limt->oo X, = -oo for m < 0.
(d) For m < 0, use (b) to compute the law of X* := max{t>0 Xt}. Note: You may use without proof that P[T < co] = limn P[Xn,b = b], where Tn,b is as in (b) {q=x:0}Jui=:9pue
