Math 236, Stochastic Differential Equations
Solutions to problem set 3
February 17, 2009
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.
Solution
First we note that
t
∧
τ
are stopping times that are bounded since
t
∧
τ
≤
t
. From Ito’s formula
we have
du
(
X
s
) =
L
u
(
X
s
) +
σ
(
X
s
)
u
(
X
s
)
dB
s
and integrating
u
(
X
t
∧
τ
) =
u
(
x
) +
integraldisplay
t
∧
τ
0
L
u
(
X
s
)
ds
+
integraldisplay
t
∧
τ
0
σ
(
X
s
)
u
x
(
X
s
)
dB
s
Assuming that
u
x
is bounded (not explicitly stated in the statement), taking expectations and
using the optional stopping theorem we have
E
x
{
u
(
X
t
∧
τ
)
}
=
u
(
x
) +
E
x
{
integraldisplay
t
∧
τ
0
L
u
(
X
s
)
ds
}
Using the hypotheses on
u
we have therefore that
E
x
{
t
∧
τ
} ≤
u
(
x
)
Since
t
∧
τ
→
τ
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 Winter '08
 Papanicolaou
 Differential Equations, Equations, Boundary value problem, optional stopping theorem, exit time, Ito’s formula

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