Description
1. W(t) is a standard Brownian motion.
(a) Let c > 0 be a constant. Show that the process defined by B(t) = cW(t/c2
) is
a standard Brownian motion.
(b) For t = n = 0, 1, . . . , show that W2
(n) − n is a discrete time martingale.
2. The nth variation of a function f, over the interval [0, T], is defined as
VT (n, f) := lim
||Π||→0
mX−1
j=0
|f(tj+1) − f(tj )|
n
in which Π = {0 = t0, t1, . . . , tm = T} is a partition of [0, T], and
||Π|| = max
0≤j≤m−1
(tj+1 − tj ).
Show that VT (1, W) = ∞ and VT (3, W) = 0, where W is a Brownian motion.
3. (a) Show that the transition probability density function for standard Brownian
motion W(t):
1
dx Pr {x < W(t + s) ≤ x + dx| W(s) = y} =
1
√
2πt
e
−
(x−y)
2
2t = f(x;t|y)
in which t, s > 0.
(b) Show that f(x;t|y) satisfies the following two linear partial differential equations:
∂f(x;t|y)
∂t =
1
2
∂
2
f(x;t|y)
∂x2
and ∂f(x;t|y)
∂t =
1
2
∂
2
f(x;t|y)
∂y2
4. Exercise 8.1: Compute d(W4
t
). Write W4
T
as an integral with respect to W plus an
integral with respect to t. Use this representation of W4
T
to show that EW4
T = 3T
2
.
Compute EW6
T using the same technique.
5. Exercise 8.2: Find an explicit expression for YT where
dYt = rdt + αYtdWt
6. Exercise 8.3: Suppose X, ∆ and Π are given by
dXt = σXtdWt
, ∆t =
∂f
∂x(t, Xt), Πt = Xt∆t
where f is some smooth function. Show that if f satisfies
∂
∂t +
1
2
σ
2x
2
∂
2
∂x2
f(t, x) = 0
for all (t, x), then Π is a martingale with respect to a filtration Ft
for all W.
7. Exercise 8.4: Suppose X is given by
dXt = µ(t, Xt)dt + σ(t, Xt)dWt
For any smooth function f define
M
f
t
:= f(t, Xt) − f(0, X0) −
Z t
0
∂
∂s + µ(s, Xs)
∂
∂x +
1
2
σ
2
(s, Xs)
∂
2
∂x2
f(s, Xs)ds.
Show that Mf
is a martingale with respect to a filtration Ft
for W.