Homework 3 (Sol)
Math 622
February 25, 2016
1. (i) Q is a Levy process so
m = E(Q(1) = b11 + b2 2 .
(ii) Consider the price model
dS(t) = S(t)dt + S(t)dM (t), S(0) = 1.
S(t) = exp ( m)t + log(1 + b1 )N1 (t) + log(1 + b2 )N2 (t) .
So K = 1, a0 = m, ai = lo

Homework 5 (Sol)
March 10, 2016
1. Let W be a Brownian motion and let cfw_F(t); t 0 be filtration for W . We
Rt
claimed in class that if Y (t) = 0 (s) dW (s), and if is a stopping time with respect
Z t
1[0, ) (s)(s) dW (s).
to cfw_F(t); t 0, then Y (t ) =

Homework 2 (Sol)
Math 622
February 19, 2016
1. Apply Itos formula:
Z t
X
u
u
(eu 1)Y (s)ds +
euN (s)s(e 1) euN (s)s(e 1)
Y (t) = 1 +
0
Z
0<st
t
0
Z
= 1+
X
(eu 1)Y (s)ds +
= 1+
u 1)
euN (s)s(e
(euN (s) 1)
0<st
t
X
(eu 1)Y (s)ds +
0
Y (s)(eu 1)N (s)
0<st
be

Homework 4 (Due 3/2/2016)
Math 622
February 25, 2016
1. Let and be stopping times with respect to a filtration cfw_F(t); t 0.
a) Show that (= mincfw_, ) is a stopping time.
b) Show that (= maxcfw_, ) is a stopping time.
2. Let X = cfw_X(t); t 0 be a stoch

Homework 4 (Sol)
Math 622
March 3, 2016
1. Let and be stopping times with respect to a filtration cfw_F(t); t 0.
a) Show that (= mincfw_, ) is a stopping time.
cfw_ > t = cfw_ > t cfw_ > t F(t),
since both and are stopping times. It follows that
cfw_ t =

Homework 1 (Due 02/10/2016)
Math 622
February 5, 2016
1. Let 0 < a < b. Let G be a c`adl`ag function of bounded variation. In the
R
R
following, the notation H(s)dG(s) will mean (0,) H(s)dG(s) as in the lecture
Note 1.
R
(i) Use the definition in Section

Math 622
Spring 2016
Midterm exam 1
3/2/16
Name (Print):
This exam contains 6 pages (including this cover page) and 6 problems. Check to see if any pages
are missing. Enter all requested information on the top of this page, and put your initials on the
to

Homework 2 (Due 02/17/2016)
Math 622
February 11, 2016
1. Let N be a Poisson process with rate and filtration F(t). Define
Y (t) := exp uN (t) t(eu 1) ,
that is Y is the exponential martingale associated with N . Use stochastic calculus
(Itos formula) for

Homework 3 (Due 2/24/2016)
Math 622
February 22, 2016
1. On a probability space (, P) let N1 , N2 be independent Poisson processes with
rate 1 , 2 and F(t) a filtration for N1 , N2 . Assume b1 > 0 > b2 > 1 and let
Q(t) := b1 N1 (t) + b2 N2 (t).
(i) Find m

Homework 5 (Due 3/9/16)
March 5, 2016
1. Let W be a Brownian motion and let cfw_F(t); t 0 be filtration for W . We claimed
Rt
in the lecture note (Theorem 4.2.16 ) that if Y (t) = 0 (s) dW (s), and if is a
Z t
1[0, ) (s)(s) dW (s).
stopping time with resp