Problem Set 3 - Mypage

ECON E724 Joon Y. Park
Economics Department Spring 2013
Problem Set 3
1. Let W be the standard Brownian motion. Show that
∫ t
∫ t
s dWs = tWt −
0
0
Ws ds.
Generalize this and derive the integration by parts formula
∫ t
∫ t
hs dWs = htWt −
for h that is of bounded variation on [0, t] for any t > 0.
0
0
0
Ws dhs
2. Let W be the standard Brownian motion, and suppose h : [0, ∞) ↦→ R is continuous.
Show that ∫ t
( ∫ t
hs dWs ∼ N 0, h 2 )
s ds
Generalize this and obtain the joint distribution of
(∫ t ∫ t
0
hs dWs,
0
0
ks dWs
where k : [0, ∞) ↦→ R is another continuous function. Use this result to find the joint
distribution of ( ∫ t )
Wt, s dWs
Note that Wt = ∫ t
0 dWs.
3. Let Bt = (Bit) be a vector Brownian motion with variance Σ = (σij). Show that
0
[Bi, Bj]t = σijt
for all i and j. In particular, [Bi, Bj]t = 0 for all t ≥ 0, if σij = 0.
4. Prove directly from the definition of Ito integrals that
∫ t
∫ t
s dMt = tMt −
0
0
)
Ms ds,
where M is a continuous martingale. Can this formula be generalized to
∫ t
∫ t
f(s) dMt = f(t)Mt − Ms df(s)
0
for any function f that is of bounded variation over any compact interval?
0

2
5. Check whether the following processes X are martingales with respect to the filtration
(Ft) generated by the standard Brownian motion W (for (a) - (c)), or the two independent
standard Brownian motions W and V (for (d)).
(a) Xt = Wt + 4t
(b) Xt = W 3 t − 3tWt
(c) Xt = t2Wt − 2 ∫ t
0 sWs ds
(d) Xt = WtVt
6. Let B be the m-dimensional standard vector Brownian motion, i.e., B is defined by
B = (B1, . . . , Bm) ′ , where Bi’s are independent standard Brownian motions. Use Ito’s
formula to write the following n-dimensional stochastic process X on the standard form
for suitable choices of u ∈ R n and v ∈ R n×m :
dXt = u(t, ω) dt + v(t, ω) dBt
(a) Xt = B 2 t , where B is 1-dimensional
(b) Xt = 2 + t + e Bt , where B is 1-dimensional
(c) Xt = (t, B 2 1t + B2 2t )′ , where B = (B1, B2) ′ is 2-dimensional
(d) Xt = (B1t + B2t + B3t, B 2 2t − B1tB3t) ′ , where B = (B1, B2, B3) ′ is 3-dimensional
7. Let W be the standard Brownian motion. Verify that the given processes solve the given
corresponding stochastic differential equations.
(a) Xt = e Wt solves
for t > 0.
(b) Xt = Wt/(1 + t) with W0 = 0 solves
for t > 0 with X0 = 0
dXt = 1
2 Xt dt + Xt dWt
dXt = − 1
1 + t Xt dt + 1
1 + t dWt
(c) Xt = sin Wt with W0 ∈ (−π/2, π/2) solves
for t < T = inf{s|Ws ∈ [−π/2, π/2]}
dXt = − 1
2 Xt dt +
√
1 − X 2 t dWt
(d) Xt = (X1t, X2t) ′ with X1t = t and X2t = etWt solves
( ) (
dX1t 1
for t > 0
dX2t
=
X2t
8. Consider the Ornstein-Uhlenbeck diffusion
)
dt +
( 0
e X1t
dXt = −Xt dt + dWt,
)
dWt

3
and compare the two methods in (a) and (b) below to obtain its quadratic variation. Which
one is wrong and why?
(a) We may derive directly from the diffusion equation
that
∫ t
Xt = X0 − Xs ds + Wt
0
[X]t = [W ]t = t,
since ∫ t
0 Xs ds is of bounded variation.
(b) We should first solve the diffusion equation to obtain
from which we may deduce
Xt = e −t ∫ t
X0 + e −(t−s) dWs,
∫ t
[X]t =
using the formula we learned from the class.
0
0
e 2(t−s) ds = 1 ( −2t
1 − e
2
)