Math 236, Stochastic Differential Equations
Problem set 2
January 22, 2009
These problems are due on
Friday January 30. 5pm
.
You can give them to me in class,
drop them in my office, or put them in my mailbox outside the mathematics department.
Problem 1
Let 0 =
t
0
< t
1
< t
2
<
· · ·
< t
N
=
T
be a partition of the interval [0
, T
] and let
B
t
, t
≥
0
,
be
the standard Brownian motion process. Show that
N

1
summationdisplay
k
=0
(
B
t
k
+1

B
t
k
)
2
converges with probability one to
T
as
N
→ ∞
and max
0
≤
k
≤
N

1
(
t
k
+1

t
k
)
→
0. That is
P
{
lim
N
→∞
N

1
summationdisplay
k
=0
(
B
t
k
+1

B
t
k
) =
T
}
= 1
.
Use the BorellCantelli lemma and the Chebychev inequality
P
{
X
 ≥
λ
} ≤
E
{
X

p
}
λ
p
with
p
= 3.
Problem 2
Let
u
(
t, x
) be a smooth (classical) solution of the terminal value PDE
u
t
(
t, x
) +
1
2
σ
2
(
x
)
u
xx
(
t, x
) +
b
(
x
)
u
x
(
t, x
) +
c
(
x
)
u
(
t, x
) = 0
,
t < T ,
x
∈
R
with terminal condition
u
(
T, x
) =
f
(
x
). Let
X
t
be the Ito process defined by
X
t
=
x
+
integraldisplay
t
0
b
(
X
s
)
ds
+
integraldisplay
t
0
σ
(
X
s
)
dB
s
,
assuming that it is well defined, with
b
(
x
),
c
(
x
) and
σ
(
x
) bounded and differentiable functions.
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 Winter '08
 Papanicolaou
 Differential Equations, Topology, Equations, Brownian Motion, Xt, Martingale

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