Math 236 Homework 5 Solutions
March 17, 2009
Problem 1.
From the definition of Gaussian, we have
p
*
(
k, x
k
, x
k
+1
) =
1
√
2
π
Δ
tσ
(
x
k
)
exp
(

(
x
k
+1

x
k

b
(
x
k
)Δ
t
)
2
2
σ
2
(
x
k
)Δ
t
)
and
p
(
k, x
k
, x
k
+1
) =
1
√
2
π
Δ
tσ
(
x
k
)
exp
(

(
x
k
+1

x
k
)
2
2
σ
2
(
x
k
)Δ
t
)
.
Then
log
M
n
=
n

1
X
k
=0
log exp
(

(
x
k
+1

x
k
)
2
+
b
2
(
x
x
)Δ
t
2

2(
x
k
+1

x
k
)
b
(
x
k
)Δ
t

(
x
k
+1

x
k
)
2
2
σ
2
(
x
k
)Δ
t
)
=
n

1
X
k
=0
b
(
x
k
)
σ
2
(
x
k
)
(
x
k
+1

x
k
)

1
2
n

1
X
k
=0
(
b
(
x
k
)
σ
(
x
k
)
)
2
Δ
t.
By direct calculation we have
E
P
[
M
n

X
0
, X
1
, . . . , X
n

1
]
=
Z
Ω
n

1
Y
k
=0
p
*
(
k, x
k
, x
k
+1
)
p
(
k, x
k
, x
k
+1
)
p
(
n

1
, x
n

1
, x
n
)d
x
n
=
Z
Ω
n

2
Y
k
=0
p
*
(
k, x
k
, x
k
+1
)
p
(
k, x
k
, x
k
+1
)
p
*
(
n

1
, x
n

1
, x
n
)d
x
n
=
M
n

1
Z
Ω
p
*
(
n

1
, x
n

1
, x
n
)d
x
n
=
M
n

1
.
Notice that in the recursive formula of
Y
n
, the value of next iterate only depends on the most current iterate.
Hence
P
(
Y
0
=
x
0
, Y
1
=
x
1
, . . . , Y
n
=
x
n
)
=
P
(
Y
0
=
x
0
)
P
(
Y
1
=
x
1

Y
0
=
x
0
)
· · ·
P
(
Y
n
=
x
n

Y
n

1
=
x
n

1
)
=
P
(
Y
0
=
x
0
)
n

1
Y
k
=0
p
*
(
k, x
k
, x
k
+1
)
.
Because
b
(
x
) =

γx
and
σ
(
x
) =
σ
0
, we have
P
(
Y
0
=
x
0
, Y
1
=
x
1
, . . . , Y
n
=
x
n
) =
P
(
Y
0
=
x
0
)
(2
π
Δ
t
)
n/
2
σ
n
0
exp

1
2
σ
2
0
Δ
t
n

1
X
k
=0
(
x
k
+1

x
k
(1

γ
Δ
t
))
2
!
.
Taking log and derivative w.r.t.
γ
, we have

1
2
σ
2
0
Δ
t
n

1
X
k
=0
2(
x
k
+1

x
k
+
x
k
γ
Δ
t
)(

x
k
Δ
t
) = 0
⇔
ˆ
γ
=

∑
n

1
k
=0
x
k
(
x
k
+1

x
k
)
Δ
t
∑
n

1
k
=0
x
2
k
.
1
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Problem 2.
As given in
Problem 1
, we know
P
{
max
0
≤
k
≤
n

Y
(
k
Δ
t
)

Y
k

> δ
} →
0
.
Thus
Y
n
converges to
Y
(
n
Δ
t
) in probability when
n
→ ∞
and
n
Δ
t
fixed. When
b
(
x
) =

γx
and
σ
(
x
) =
σ
,
i.e.
Y
(
t
) satisfies
d
Y
(
t
) =

γY
(
t
)d
t
+
σ
d
B
(
t
)
,
we know
Y
(
t
) is an OU process. When Δ
t
→
0 and
n
Δ
t
=
t
, we have
n

1
X
k
=0
y
k
(
y
k
+1

y
k
)
≈
Z
t
0
Y
(
s
)d
Y
(
s
)
,
Δ
t
n

1
X
k
=0
y
2
k
≈
Z
t
0
Y
2
(
s
)d
s.
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 Winter '08
 Papanicolaou
 Math, Differential Equations, Equations, Trigraph, xk, Lχ, χy, σχy

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