Math 236, Stochastic Differential Equations
Problem set 3
February 5, 2009
These problems are due on
Friday February 13
. 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
X
t
be the onedimensional diffusion process satisfying the Ito SDE
dX
t
=
b
(
X
t
)
dt
+
σ
(
X
t
)
dB
t
,
X
0
=
x
where
b
(
x
) and
σ
(
x
) satisfy the Ito conditions (linear growth bound and Lipschitz condition). Let
x
∈
(
α, β
) and let
τ
=
τ
x
= inf
{
t >
0
:
X
t
∈
[
α, β
]
c
}
be the exit time from the interval [
α, β
]
starting from
x
in its interior, and let
L
be the generator of the process
L
=
1
2
σ
2
(
x
)
∂
2
∂x
2
+
b
(
x
)
∂
∂x
Show using Ito’s formula and the optional stopping theorem that if there is a nonnegative and
bounded function
u
(
x
) for
x
∈
[
α, β
] such that
L
u
(
x
)
≤ 
1, then
E
x
{
τ
}
<
∞
and so
τ
x
<
∞
with
probability one.
Problem 2
With the notation of Problem 1, show that if the solution of the boundary value problem
L
u
(
x
) =

1 for
x
∈
(
α, β
) with
u
(
α
) =
u
(
β
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '08
 Papanicolaou
 Differential Equations, Equations, Boundary value problem

Click to edit the document details