We divide our derivation into two cases. In the first case,
γ
<
1
so that the utility function is proportional to a power function. It refers to a power utility. In the second case, we take the limit of
γ
→
1
on the CRRA utility. As the limit is the logarithmic function, it refers to the logarithmic utility or Bernoulli's utility function.
3.1. Power Utility
In our research problem, the optimal decision is not affected by adding a real constant to the objective function. The power utility maximization problem can then be reduced to
(14)
max
E
[
Y
(
T
)
1
-
γ
1
-
γ
]
,
where the insurer’s wealth follows the SDE in (11).
Theorem 4.
Under Assumptions 1 and 2, the research problem (13) with the power utility (14) has the optimal solution
(
investment policy
)
,
(15)
u
s
*
(
t
,
Y
(
t
)
,
r
(
t
)
)
=
1
γ
(
Σ
s
′
(
t
)
)
-
1
Λ
(
t
)
Y
(
t
)
;
u
B
*
(
t
,
Y
(
t
)
,
r
(
t
)
)
=
r
(
t
)
γ
σ
B
(
t
)
(
λ
r
(
t
)
-
Λ
′
(
t
)
Σ
s
(
t
)
-
1
Σ
r
(
t
)
-
K
(
t
,
T
)
σ
r
(
t
)
(
λ
r
(
t
)
-
Λ
′
(
t
)
Σ
s
(
t
)
-
1
Σ
r
(
t
)
)
Y
(
t
)
,
and the optimal value of the objective function
(16)
E
[
Y
(
T
)
1
-
γ
1
-
γ
|
G
0
]
|
u
=
u
*
=
Y
0
1
-
γ
1
-
γ
exp
[
K
(
0
,
T
)
r
(
0
)
+
M
′
(
0
,
T
)
]
,
where
K
(
t
,
T
)
and
M
(
t
,
T
)
satisfy the system of ordinary differential equations (ODE):
(17)
K
˙
(
t
,
T
)
-
(
b
(
t
)
+
1
-
γ
γ
σ
r
(
t
)
λ
r
(
t
)
)
K
(
t
,
T
)
+
σ
r
(
t
)
2
2
γ
K
(
t
,
T
)
2
+
(
1
-
γ
+
1
-
γ
2
γ
λ
r
(
t
)
2
)
=
0
,
K
(
T
,
T
)
=
0
;
(18)
M
˙
(
t
,
T
)
+
a
(
t
)
K
(
t
,
T
)
+
E
[
(
e
-
z
(
1
-
γ
)
-
1
)
μ
(
t
)
]
-
1
-
γ
2
γ
∥
Λ
(
t
)
∥
2
=
0
,
M
(
T
,
T
)
=
0
.
Proof.
The proof is based on the classic HJB framework. Let
(19)
V
(
t
,
y
,
r
)
=
sup
u
∈
Π
E
[
U
(
Y
u
(
T
)
)
∣
G
t
]
.
For a fixed terminal time
T
, the corresponding HJB equation is
(20)
V
t
+
V
r
′
(
a
-
b
r
)
+
1
2
tr
(
V
r
r
σ
r
2
r
)
+
E
[
(
V
(
t
,
y
-
y
(
1
-
e
-
z
)
,
r
)
-
V
(
t
,
y
,
r
)
)
μ
]
+
sup
{
1
2
V
y
(
r
y
+
u
s
′
(
Σ
s
Λ
+
Σ
r
λ
r
r
)
+
u
B
σ
B
λ
r
r
)
u
-
V
r
y
σ
r
r
(
u
s
′
Σ
r
r
+
u
B
σ
B
)
+
1
2
V
y
y
(
u
s
′
Σ
s
Σ
s
′
u
s
+
u
s
′
Σ
r
Σ
r
′
u
s
r
+
u
B
2
σ
B
2
+
2
u
s
′
Σ
r
r
u
B
σ
B
)
1
2
}
=
0
,
with
V
(
T
,
y
,
r
)
=
y
1
-
γ
/
(
1
-
γ
)
. Thus, the optimal feedback control,
u
*
, maximizes
(21)
V
y
u
′
(
Σ
s
Λ
+
Σ
r
λ
r
r
σ
B
λ
r
r
)
-
V
r
y
σ
r
r
u
′
(
Σ
r
r
σ
B
)
+
1
2
V
y
y
u
′
(
Σ
s
Σ
s
′
+
Σ
s
Σ
s
′
r
Σ
r
σ
B
r
Σ
r
′
σ
B
r
σ
B
2
)
u
,
where
u
=
(
u
s
u
B
)
′
. If
V
y
y
<
0
, differentiating (21) with respect to
u
and setting the differential to zero results in
(22)
u
*
=
Σ
-
1
(
V
r
y
V
y
y
σ
r
r
(
Σ
r
r
σ
B
)
-
V
y
V
y
y
(
Σ
s
Λ
+
Σ
r
λ
r
r
σ
B
λ
r
r
)
)
,
where
Σ
=
(
Σ
s
Σ
s
′
+
Σ
s
Σ
s
′
r
Σ
r
σ
B
r
Σ
r
′
σ
B
r
σ
B
2
)
. Otherwise, if
V
y
y
≥
0
, then the optimization has no solution.
Note that
Σ
-
1
can be simplified by matrix inversion lemma (or called Sherman-Morrison-Woodbury formula). Hence, we have
(23)
Σ
-
1
=
(
(
Σ
s
Σ
s
′
)
-
1
-
r
σ
B
(
Σ
s
Σ
s
′
)
-
1
Σ
r
-
r
σ
B
Σ
r
′
(
Σ
s
Σ
s
′
)
-
1
1
+
r
Σ
r
′
(
Σ
s
Σ
s
′
)
-
1
Σ
r
σ
B
2
)
.
Therefore, (22) becomes
(24)
u
*
=
(
u
s
*
u
B
*
)
=
(
-
V
y
V
y
y
(
Σ
s
′
)
-
1
Λ
r
σ
B
[
V
r
y
V
y
y
σ
r
-
V
y
V
y
y
(
λ
r
-
Λ
′
Σ
s
-
1
Σ
r
)
]
)
.
Substituting the
u
*
into the HJB equation (20), the PIDE of
V
becomes
(25)
V
t
+
V
r
(
a
-
b
r
)
+
1
2
V
r
r
σ
r
2
r
+
V
y
r
y
+
E
[
(
V
(
t
,
y
-
y
(
1
-
e
-
z
)
,
r
)
-
V
)
μ
]
-
1
2
V
y
y
[
V
r
y
V
y
y
σ
r
r
(
Σ
r
r
σ
B
)
-
V
y
V
y
y
(
Σ
s
Λ
+
Σ
r
λ
r
r
σ
B
λ
r
r
)
]
′
×
Σ
-
1
[
V
r
y
V
y
y
σ
r
r
(
Σ
r
r
σ
B
)
-
V
y
V
y
y
(
Σ
s
Λ
+
Σ
r
λ
r
r
σ
B
λ
r
r
)
]
=
0
,
with terminal condition
V
(
T
,
y
,
r
)
=
U
(
y
)
. As
U
(
y
)
=
y
1
-
γ
/
(
1
-
γ
)
, consider an exponential affine form for
V
:
(26)
V
(
t
,
y
,
r
)
=
y
1
-
γ
1
-
γ
exp
[
K
(
t
,
T
)
r
+
M
(
t
,
T
)
]
,
where
K
and
M
are deterministic functions of
t
and satisfy the ODEs (17) and (18), respectively. Note that the ODEs (17) can be regarded as a Riccati equation. Clearly, the terminal value of the function in (26) satisfies the terminal condition in (25) and
V
y
y
<
0
. Taking partial derivatives to the affine form
V
with respect to
t
,
y
, and
r
, we have
(27)
V
t
=
(
K
˙
r
+
M
˙
)
V
;
V
r
=
K
V
;
V
y
=
1
-
γ
y
V
;
V
r
r
=
K
2
V
;
V
y
y
=
-
γ
(
1
-
γ
)
y
2
V
;
V
r
y
=
1
-
γ
y
K
V
;
E
[
(
V
(
t
,
y
-
y
(
1
-
e
-
z
)
,
r
)
-
V
)
μ
]
=
E
[
(
e
-
z
(
1
-
γ
)
-
1
)
μ
]
V
.
After substituting these expressions into the left-hand side of (25), simple but tedious calculations easily verify that the proposed solution form satisfies the PIDE in (25). Thus, the solution form in (26) is actually a solution of the PDE in (25). As the value function is twice continuously differentiable and all of the parameters are uniformly bounded and predictable, the classical verification theorems of [12] (III, Theorem 8.1) confirm that the proposed affine form of value function in (26) and the control in (15) are the optimal value function and optimal feedback control, respectively.
Theorem 4 asserts that if we are able to solve the system of ODEs (17) and (18), then both optimal investment policy
u
*
and the optimal function value are efficiently computed. In fact, the solution to (18) is simple given the solution of (17) because (18) is linear ODE. However, ODE (17) is a Riccati differential equation (RDE). Using Radon’s lemma (c.f. Theorem
3.1
.
1
of the book [13]), the matrix RDE (17) can be solved systematically.
Proposition 5.
If
K
(
t
,
T
)
satisfies the matrix RDE (17) and
M
(
t
,
T
)
satisfies the linear ODE in (18), then the solution is explicitly obtained as follows.
(
1
)
K
(
τ
)
:
=
K
(
t
,
T
)
=
R
2
(
τ
)
R
1
-
1
(
τ
)
, where
τ
=
T
-
t
, and
R
=
(
R
1
(
τ
)
′
R
2
(
τ
)
′
)
′
is the solution of the linear system of ODEs in the interval
[
0
,
T
]
:
(28)
d
R
d
τ
=
(
1
2
(
b
(
t
)
+
1
-
γ
γ
σ
r
(
t
)
λ
r
(
t
)
)
-
σ
r
(
t
)
2
2
γ
1
-
γ
+
1
-
γ
2
γ
λ
r
(
t
)
2
-
1
2
(
b
(
t
)
+
1
-
γ
γ
σ
r
(
t
)
λ
r
(
t
)
)
)
R
,
R
(
0
)
=
(
I
m
0
m
×
m
)
.
In particular, if
b
(
t
)
,
σ
r
(
t
)
, and
λ
r
(
t
)
are constants, then
(29)
R
(
τ
)
=
exp
[
(
1
2
(
b
+
1
-
γ
γ
σ
r
λ
r
)
-
σ
r
2
2
γ
1
-
γ
+
1
-
γ
2
γ
λ
r
2
-
1
2
(
b
+
1
-
γ
γ
σ
r
λ
r
)
)
τ
]
(
I
m
0
m
×
m
)
.
And,
(
2
)
(30)
M
(
t
,
T
)
=
∫
t
T
[
[
-
1
-
γ
2
γ
∥
Λ
(
s
)
∥
2
]
d
s
]
a
(
s
)
K
(
s
,
T
)
+
E
[
(
e
-
z
(
1
-
γ
)
-
1
)
μ
(
s
)
]
-
1
-
γ
2
γ
∥
Λ
(
s
)
∥
2
]
d
s
,
where
E
[
e
-
z
(
1
-
γ
)
]
is the moment generating function of
z
.
Proof.
The solution of the matrix Riccati differential equation (17) can be solved by the Radon lemma, a proof of which can be found in Theorem
3.1
.
1
of [13]. The Radon lemma immediately gives the explicit expression for
K
(
t
,
T
)
.
M
(
t
,
T
)
is obtained by a simple and direct integration. Hence, the results follow.
3.3. Verification Theorem
The following two propositions together serve as a verification theorem for the solution of the HJB equation in (20). The results and proofs are classical. We adopt the framework of [10]. For smoothening the proofs of the propositions, a notation is introduced as follows:
(41)
L
u
J
=
∂
J
∂
t
+
∂
J
∂
r
(
a
-
b
r
)
+
∂
J
∂
y
(
r
y
+
u
′
(
Σ
s
Λ
+
Σ
r
λ
r
r
λ
r
r
)
)
+
1
2
∂
2
J
∂
r
2
σ
r
2
r
+
1
2
∂
2
J
∂
y
2
u
′
(
Σ
s
Σ
s
′
+
Σ
s
Σ
s
′
r
Σ
r
σ
B
r
Σ
r
′
σ
B
r
σ
B
2
)
u
-
∂
2
J
∂
r
∂
y
σ
r
r
u
′
(
Σ
r
r
σ
B
)
+
E
[
(
J
(
t
,
y
-
y
(
1
-
e
-
z
)
,
r
)
-
J
(
t
,
y
,
r
)
)
μ
]
.
Clearly, the HJB equation in (20) can be rewritten as
sup
u
L
u
V
=
0
.
Proposition 7.
It is assumed that
J
∈
C
2
(
[
0
,
T
]
×
R
+
2
)
is a nonnegative function such that
L
u
J
≤
0
,
J
(
T
,
y
,
r
)
=
U
(
y
)
, and
J
(
t
,
0
,
r
)
=
0
for every admissible control
u
. Then
(42)
J
(
t
,
y
,
r
)
≥
E
[
U
(
Y
u
(
T
)
)
∣
G
t
]
.
Proof.
It is assumed that
u
is an admissible control, and let
{
τ
n
}
n
be a localizing sequence of stopping times for the local semimartingale
(
Y
u
(
s
)
,
r
(
s
)
)
starting with
(
y
,
r
)
at time
t
. Applying Itô’s lemma to
J
with respect to (11), we have
(43)
d
J
=
{
∂
2
J
∂
r
∂
y
σ
r
r
u
′
(
Σ
r
r
σ
B
)
}
∂
J
∂
t
+
∂
J
∂
r
(
a
-
b
r
)
+
∂
J
∂
y
(
r
y
+
u
′
(
Σ
s
Λ
+
Σ
r
λ
r
r
λ
r
r
)
)
+
1
2
∂
2
J
∂
r
2
σ
r
2
r
+
1
2
∂
2
J
∂
y
2
u
′
(
Σ
s
Σ
s
′
+
Σ
s
Σ
s
′
r
Σ
r
σ
B
r
Σ
r
′
σ
B
r
σ
B
2
)
u
-
∂
2
J
∂
r
∂
y
σ
r
r
u
′
(
Σ
r
r
σ
B
)
}
d
t
+
{
∂
J
∂
y
u
′
(
Σ
s
Σ
r
r
0
1
×
n
σ
B
)
-
∂
J
∂
r
(
0
1
×
n
σ
r
r
)
}
d
W
t
+
(
J
(
t
,
y
-
y
(
1
-
e
-
z
)
,
r
)
-
J
(
t
,
y
,
r
)
)
d
N
t
=
{
-
∂
2
J
∂
r
∂
y
σ
r
r
u
′
(
Σ
r
r
σ
B
)
(
r
y
+
u
′
(
Σ
s
Λ
+
Σ
r
λ
r
r
λ
r
r
)
)
}
∂
J
∂
t
+
∂
J
∂
r
(
a
-
b
r
)
+
∂
J
∂
y
(
r
y
+
u
′
(
Σ
s
Λ
+
Σ
r
λ
r
r
λ
r
r
)
)
+
1
2
∂
2
J
∂
r
2
σ
r
2
r
+
1
2
∂
2
J
∂
y
2
u
′
(
Σ
s
Σ
s
′
+
Σ
s
Σ
s
′
r
Σ
r
σ
B
r
Σ
r
′
σ
B
r
σ
B
2
)
u
-
∂
2
J
∂
r
∂
y
σ
r
r
u
′
(
Σ
r
r
σ
B
)
(
r
y
+
u
′
(
Σ
s
Λ
+
Σ
r
λ
r
r
λ
r
r
)
)
}
d
t
+
(
J
(
t
,
y
-
y
(
1
-
e
-
z
)
,
r
)
-
J
(
t
,
y
,
r
)
)
μ
(
t
)
d
t
+
{
∂
J
∂
y
u
′
(
Σ
s
Σ
r
r
0
1
×
n
σ
B
)
-
∂
J
∂
r
(
0
1
×
n
σ
r
r
)
}
d
W
t
+
(
J
(
t
,
y
-
y
(
1
-
e
-
z
)
,
r
)
-
J
(
t
,
y
,
r
)
)
d
M
t
.
Hence, we have
(44)
E
[
J
(
T
∧
τ
n
,
Y
u
(
T
∧
τ
n
)
,
r
(
T
∧
τ
n
)
)
∣
G
t
]
=
J
(
t
,
y
,
r
)
+
∫
t
T
∧
τ
n
L
u
J
(
s
,
Y
u
(
s
)
,
r
(
s
)
)
d
s
≤
J
(
t
,
y
,
r
)
.
Because of the nonnegativity of
J
,
{
J
(
T
∧
τ
n
,
Y
u
(
T
∧
τ
n
)
,
r
(
T
∧
τ
n
)
)
}
n
is a sequence of nonnegative measurable functions. Since
(45)
lim
n
→
∞
J
(
T
∧
τ
n
,
Y
u
(
T
∧
τ
n
)
,
r
(
T
∧
τ
n
)
)
=
J
(
T
,
Y
u
(
T
)
,
r
(
T
)
)
=
U
(
Y
u
(
T
)
)
,
a
.
s
.
,
Fatou’s lemma yields
(46)
E
[
U
(
Y
u
(
T
)
)
∣
G
t
]
=
E
[
lim
inf
n
→
∞
J
(
T
∧
τ
n
,
Y
u
(
T
∧
τ
n
)
,
r
(
T
∧
τ
n
)
)
∣
G
t
]
≤
lim
inf
n
→
∞
E
[
J
(
T
∧
τ
n
,
Y
u
(
T
∧
τ
n
)
,
r
(
T
∧
τ
n
)
)
∣
G
t
]
≤
J
(
t
,
y
,
r
)
.
Proposition 8.
Let
J
∈
C
2
(
[
0
,
T
]
×
R
+
2
)
be a nonnegative function such that the family of random variables
{
J
(
τ
,
Y
u
*
(
τ
)
,
r
(
τ
)
)
}
τ
is uniformly integrable, where
u
*
is an admissible control with the property
L
u
*
J
=
0
and
τ
∈
[
t
,
T
]
is a stopping time for the process
(
Y
u
*
(
s
)
,
r
(
s
)
)
starting with
(
y
,
r
)
at time
t
. If, furthermore,
J
(
T
,
y
,
r
)
=
U
(
y
)
,
J
(
t
,
0
,
r
)
=
0
, and
L
u
J
≤
0
for all admissible controls
u
, then
(47)
J
(
t
,
y
,
r
)
=
V
(
t
,
y
,
r
)
∀
(
t
,
y
,
r
)
∈
[
0
,
T
]
×
R
+
2
.
Proof.
Thanks to the uniform integrability of the family
{
J
τ
,
(
Y
u
*
(
τ
)
,
r
(
τ
)
)
}
τ
, we have
(48)
E
[
U
(
Y
u
*
(
T
)
)
∣
G
t
]
=
E
[
lim
n
→
∞
J
(
T
∧
τ
n
,
Y
u
*
(
T
∧
τ
n
)
,
r
(
T
∧
τ
n
)
)
∣
G
t
]
=
lim
n
→
∞
E
[
J
(
T
∧
τ
n
,
Y
u
*
(
T
∧
τ
n
)
,
r
(
T
∧
τ
n
)
)
∣
G
t
]
≤
J
(
t
,
y
,
r
)
.
Furthermore,
L
u
*
J
=
0
induce that
E
[
U
(
Y
u
*
(
T
)
)
∣
G
t
]
=
J
(
t
,
y
,
r
)
. Hence, by Proposition 7,
J
(
t
,
y
,
r
)
=
V
(
t
,
y
,
r
)
for all
(
t
,
y
,
r
)
∈
[
0
,
T
]
×
R
+
2
.
The above propositions are classical and valid for a class of positive utility functions including the CRRA utility. To verify the uniform integrability of
{
J
(
τ
,
Y
u
*
(
τ
)
,
r
(
τ
)
)
}
τ
, we need to calculate
E
[
J
(
τ
,
Y
u
*
(
τ
)
,
r
(
τ
)
)
∣
G
t
]
, where
(49)
J
(
τ
,
Y
(
τ
)
,
r
(
τ
)
)
=
{
V
(
τ
,
Y
(
τ
)
,
r
(
τ
)
)
when
U
(
y
)
=
y
1
-
γ
1
-
γ
;
ln
Y
(
τ
)
+
∫
τ
T
∥
Λ
(
s
)
∥
2
2
d
s
-
E
[
z
]
∫
τ
T
E
[
μ
(
s
)
]
d
s
+
∫
τ
T
(
+
∫
τ
s
e
-
∫
η
s
b
(
θ
)
d
θ
a
(
η
)
d
η
)
e
-
∫
τ
s
b
(
η
)
d
η
r
(
τ
)
+
∫
τ
s
e
-
∫
η
s
b
(
θ
)
d
θ
a
(
η
)
d
η
)
×
(
1
-
λ
r
(
s
)
2
2
)
d
s
when
U
(
y
)
=
ln
y
;
V
(
t
,
Y
(
t
)
,
r
(
τ
)
)
is given in (26). The determination of the positivity of
K
(the solution of the Riccati differential equation (17)) will be discussed. The following proposition is a comparison theorem for the solutions of standard Riccati differential equations. The details can be found in [14].
Proposition 9.
For
i
=
1,2
, let
K
i
be the solution of
(50)
K
˙
i
=
-
A
i
′
(
t
)
K
i
-
K
i
A
i
(
t
)
-
Q
i
(
t
)
+
K
i
S
i
(
t
)
K
i
on some interval
I
. If for some
t
f
∈
I
,
K
1
(
t
f
)
≤
K
2
(
t
f
)
and if
(51)
(
Q
2
A
2
′
A
2
-
S
2
)
(
t
)
-
(
Q
1
A
1
′
A
1
-
S
1
)
(
t
)
≥
0
f
o
r
t
∈
I
,
where
0
is a matrix with zero valued entries, then
K
1
(
t
)
≤
K
2
(
t
)
for all
t
∈
I
∩
(
-
∞
,
t
f
]
.
Lemma 10.
The solution of ODE (17),
K
(
t
,
T
)
, is a nonnegative function of time
t
for all
t
∈
[
0
,
T
]
.
Proof.
Consider
A
1
=
A
2
=
-
(
1
/
2
)
(
b
+
(
(
1
-
γ
)
/
γ
)
σ
r
λ
r
)
,
S
1
=
S
2
=
-
σ
r
2
/
2
γ
<
0
,
Q
1
=
0
, and
Q
2
=
(
1
-
γ
+
(
(
1
-
γ
)
/
2
γ
)
λ
r
2
)
>
0
. It is obvious that
0
m
×
m
is the solution of
(52)
K
˙
1
=
-
A
1
′
(
t
)
K
1
-
K
1
A
1
(
t
)
-
Q
1
(
t
)
+
K
1
S
1
(
t
)
K
1
,
K
1
(
T
)
=
0
;
and the solution of the ODE (17),
K
(
t
,
T
)
, is the solution of
(53)
K
˙
2
=
-
A
2
′
(
t
)
K
2
-
K
2
A
2
(
t
)
-
Q
2
(
t
)
+
K
2
S
2
(
t
)
K
2
,
K
2
(
T
)
=
0
.
Therefore, we have
(54)
(
Q
2
A
2
′
A
2
-
S
2
)
(
t
)
-
(
Q
1
A
1
′
A
1
-
S
1
)
(
t
)
=
(
1
-
γ
+
1
-
γ
2
γ
λ
r
2
0
0
0
)
(
t
)
≥
0
2
×
2
and
K
2
(
T
)
≥
K
1
(
T
)
. Applying Proposition 9,
K
(
t
,
T
)
=
K
2
(
t
)
≥
0
for all
t
∈
[
0
,
T
]
, which means that the solution of the matrix RDE (17),
K
(
t
,
T
)
, is a semipositive definite matrix for all
t
∈
[
0
,
T
]
.
Next step is a calculation of the expected value of
J
(
τ
,
Y
u
*
(
τ
)
,
r
(
τ
)
)
at filtration
G
t
, where
0
≤
t
≤
τ
.
Lemma 11.
Given
0
≤
t
≤
τ
≤
T
and using the notations in Theorem 4,
(55)
E
[
V
(
τ
,
Y
u
*
(
τ
)
,
r
(
τ
)
)
∣
G
t
]
=
E
[
(
Y
u
*
(
τ
)
)
1
-
γ
1
-
γ
e
K
(
τ
,
T
)
r
(
τ
)
+
M
(
τ
,
T
)
∣
G
t
]
=
(
Y
(
t
)
)
1
-
γ
ψ
μ
*
(
t
,
τ
)
(
ψ
z
(
1
-
γ
)
-
1
)
1
-
γ
e
K
~
(
t
,
τ
)
r
(
t
)
+
M
~
(
t
,
τ
)
,
where
K
~
(
t
,
τ
)
and
M
~
(
t
,
τ
)
satisfy the system of ordinary differential equations (ODEs):
(56)
K
~
˙
-
(
b
+
1
-
γ
γ
(
λ
r
σ
r
-
K
σ
r
2
)
)
+
σ
r
2
2
K
~
2
+
1
-
γ
+
1
-
γ
2
γ
(
λ
r
2
-
K
2
σ
r
2
)
=
0
,
K
~
(
τ
,
τ
)
=
K
(
τ
,
T
)
;
(57)
M
~
˙
+
K
~
a
+
1
-
γ
2
γ
∥
Λ
∥
2
=
0
,
M
~
(
τ
,
τ
)
=
M
(
τ
,
T
)
,
ψ
Z
(
t
)
is a moment generating function of random variable
Z
and
μ
*
(
t
,
τ
)
=
∫
t
τ
μ
(
s
)
d
s
.
Proof.
Let
(58)
v
~
(
t
,
Y
,
r
)
=
Y
1
-
γ
ψ
μ
*
(
t
,
τ
)
(
ψ
z
(
1
-
γ
)
-
1
)
1
-
γ
e
K
~
(
t
,
τ
)
r
+
M
~
(
t
,
τ
)
,
where
K
~
and
M
~
are the solution of the system of ODEs (56) and (57), respectively;
r
follows the dynamic:
(59)
d
r
(
t
)
=
(
a
-
b
r
)
d
t
-
σ
r
r
d
W
t
r
and
Y
*
follows the dynamic:
(60)
d
Y
Y
=
r
*
d
t
+
σ
*
d
W
t
-
(
1
-
e
-
z
)
d
N
t
,
r
*
=
r
(
1
+
1
+
λ
r
2
-
λ
r
K
σ
r
γ
)
+
∥
Λ
∥
2
γ
,
σ
*
=
1
γ
(
Λ
′
r
(
λ
r
-
K
σ
r
)
)
;
ψ
Z
(
t
)
is a moment generating function of random variable
Z
and
μ
*
(
t
,
τ
)
=
∫
t
τ
μ
(
s
)
d
s
. Clearly,
(61)
v
~
(
τ
,
Y
,
r
)
=
Y
1
-
γ
1
-
γ
e
K
(
τ
,
T
)
r
+
M
(
τ
,
T
)
=
E
[
Y
(
τ
)
1
-
γ
1
-
γ
e
K
(
τ
,
T
)
r
(
τ
)
+
M
(
τ
,
T
)
∣
G
τ
]
=
E
[
V
(
τ
,
Y
u
*
(
τ
)
,
r
(
τ
)
)
∣
G
τ
]
.
Next,
(62)
v
~
t
+
v
~
r
(
a
-
b
r
)
+
1
2
v
~
r
r
σ
r
2
r
+
E
[
(
v
~
(
t
,
Y
-
Y
(
1
-
e
-
z
)
,
r
)
-
v
~
(
t
,
Y
,
r
)
)
μ
]
+
v
~
Y
(
r
*
Y
)
+
1
2
v
~
Y
Y
σ
*
σ
*
′
Y
2
-
v
~
r
Y
(
r
γ
σ
r
(
λ
r
-
K
σ
r
)
)
Y
would be calculated. Taking partial derivatives to
v
~
with respect to
t
,
Y
, and
r
, we have
(63)
v
~
t
=
(
-
μ
(
ψ
z
(
1
-
γ
)
-
1
)
+
K
~
˙
r
+
M
~
˙
)
v
~
=
(
-
E
[
μ
(
e
-
(
1
-
γ
)
z
-
1
)
]
+
K
~
˙
′
r
+
M
˙
)
v
~
;
v
~
r
=
K
~
v
~
;
v
~
r
r
=
K
~
2
v
~
;
v
~
y
=
1
-
γ
y
v
~
;
v
~
y
y
=
-
γ
(
1
-
γ
)
y
2
v
~
;
v
~
r
y
=
1
-
γ
y
K
~
v
~
;
E
[
(
v
~
(
t
,
y
-
y
(
1
-
e
-
z
)
,
r
)
-
v
~
)
μ
]
=
E
[
(
e
-
z
(
1
-
γ
)
-
1
)
μ
]
v
~
.
Substituting these expressions into (62) yields
(64)
v
~
t
+
v
~
r
(
a
-
b
r
)
+
1
2
v
~
r
r
σ
r
2
r
+
E
[
(
v
~
(
t
,
Y
-
Y
(
1
-
e
-
z
)
,
r
)
-
v
~
(
t
,
Y
,
r
)
)
μ
]
+
v
~
Y
(
r
*
Y
)
+
1
2
v
~
Y
Y
σ
*
σ
*
′
Y
2
-
v
~
r
Y
(
r
γ
σ
r
(
λ
r
-
K
σ
r
)
)
Y
=
v
~
{
[
+
σ
r
2
2
K
~
2
+
1
-
γ
+
1
-
γ
2
γ
(
λ
r
2
-
K
2
σ
r
2
)
]
K
~
˙
-
(
b
+
1
-
γ
γ
(
λ
r
σ
r
-
K
σ
r
2
)
)
+
σ
r
2
2
K
~
2
+
1
-
γ
+
1
-
γ
2
γ
(
λ
r
2
-
K
2
σ
r
2
)
]
r
+
M
~
˙
+
K
~
a
+
1
-
γ
2
γ
∥
Λ
∥
2
{
[
+
σ
r
2
2
K
~
2
+
1
-
γ
+
1
-
γ
2
γ
(
λ
r
2
-
K
2
σ
r
2
)
]
K
~
-
(
b
+
1
-
γ
γ
(
λ
r
σ
r
-
K
σ
r
2
)
)
}
=
0
.
Combining the equalities (61) and (64),
v
~
(
t
,
Y
,
r
)
=
E
[
V
(
τ
,
Y
u
*
(
τ
)
,
r
(
τ
)
)
∣
G
t
]
is proved.
To show the uniform integrability of
{
V
(
τ
,
Y
u
*
(
τ
)
,
r
(
τ
)
)
}
τ
is equivalent to showing the boundedness of the function value
v
~
(
t
,
Y
,
r
)
by Lemma 11. Hence, the boundedness of the solution of ODE in (56),
K
~
(
t
,
τ
)
, induces the boundedness of function value
v
~
(
t
,
Y
,
r
)
. The following proposition (c.f. Theorem 4.3 of [15]) is useful for showing the boundedness of
K
~
(
t
,
τ
)
.
Lemma 12.
Given
0
≤
t
≤
τ
≤
T
and using the notations in Theorem 4 and Lemma 11,
(65)
K
~
(
t
,
τ
)
≤
K
(
t
,
T
)
,
for any stopping times
τ
∈
[
0
,
T
]
and
t
∈
[
0
,
τ
]
.
Proof.
Consider a stopping time
τ
such that
τ
∈
[
0
,
T
]
and
t
∈
[
0
,
τ
]
. By choosing
(66)
A
1
=
-
1
2
(
b
+
1
-
γ
γ
(
λ
r
σ
r
-
K
σ
r
2
)
)
;
A
2
=
-
1
2
(
b
+
1
-
γ
γ
σ
r
λ
r
)
;
S
1
=
-
σ
r
2
2
;
S
2
=
-
σ
r
2
2
γ
;
Q
1
=
1
-
γ
+
1
-
γ
2
γ
(
λ
r
2
-
K
2
σ
r
2
)
;
Q
2
=
1
-
γ
+
1
-
γ
2
γ
λ
r
2
,
it is obvious that
K
~
(
t
,
τ
)
is the solution of
(67)
K
˙
1
=
-
A
1
′
(
t
)
K
1
-
K
1
A
1
(
t
)
-
Q
1
(
t
)
+
K
1
S
1
(
t
)
K
1
,
K
˙
1
=
-
A
1
′
(
t
)
K
1
-
k
k
K
1
A
1
(
t
)
-
K
1
(
τ
)
=
K
~
(
τ
,
τ
)
;
and the solution of the matrix RDE (17),
K
(
t
,
T
)
, is the solution of
(68)
K
˙
2
=
-
A
2
′
(
t
)
K
2
-
K
2
A
2
(
t
)
-
Q
2
(
t
)
+
K
2
S
2
(
t
)
K
2
,
K
˙
2
=
-
A
2
′
(
t
)
K
2
-
K
2
A
2
(
t
)
-
Q
2
K
2
(
τ
)
=
K
(
τ
,
T
)
.
Therefore, we have
(69)
(
Q
2
A
2
′
A
2
-
S
2
)
(
t
)
-
(
Q
1
A
1
′
A
1
-
S
1
)
(
t
)
=
1
-
γ
2
γ
(
K
2
σ
r
2
-
K
σ
r
2
-
K
σ
r
2
σ
r
2
)
(
t
)
.
Clearly,
K
2
σ
r
2
and
(
1
-
γ
)
/
2
γ
are greater than zero; and
(70)
det
(
K
2
σ
r
2
-
K
σ
r
2
-
K
σ
r
2
σ
r
2
)
=
0
.
Hence, the matrix in (69),
(
(
1
-
γ
)
/
2
γ
)
(
K
2
σ
r
2
-
K
σ
r
2
-
K
σ
r
2
σ
r
2
)
(
t
)
, is semipositive definite. Combining the fact that
K
(
τ
,
T
)
=
K
2
(
τ
)
≥
K
1
(
τ
)
=
K
~
(
τ
,
τ
)
and applying Proposition 9,
K
~
(
t
,
τ
)
≥
K
(
t
,
T
)
for all
t
∈
[
0
,
τ
]
.
Lemma 13.
Given
0
≤
t
≤
τ
≤
T
and using the notations in Theorem 4 and Lemma 11,
(71)
k
(
t
,
τ
)
≤
K
~
(
t
,
τ
)
,
for any stopping times
τ
∈
[
0
,
T
]
and
t
∈
[
0
,
τ
]
, where
k
(
t
,
τ
)
is the non-blow-up solution of the following RDE:
(72)
k
˙
-
(
b
+
1
-
γ
γ
(
λ
r
σ
r
-
K
σ
r
2
)
)
+
σ
r
2
2
k
2
-
1
-
γ
2
γ
K
2
σ
r
2
=
0
,
k
(
τ
,
τ
)
=
K
(
τ
,
T
)
.
Proof.
Consider a stopping time
τ
such that
τ
∈
[
0
,
T
]
and
t
∈
[
0
,
τ
]
. By choosing
(73)
A
1
=
-
1
2
(
b
+
1
-
γ
γ
(
λ
r
σ
r
-
K
σ
r
2
)
)
;
A
2
=
-
1
2
(
b
+
1
-
γ
γ
(
λ
r
σ
r
-
K
σ
r
2
)
)
;
S
1
=
-
σ
r
2
2
;
S
2
=
-
σ
r
2
2
;
Q
1
=
-
1
-
γ
2
γ
K
2
σ
r
2
;
Q
2
=
1
-
γ
+
1
-
γ
2
γ
(
λ
r
2
-
K
2
σ
r
2
)
,
it is obvious that
k
(
t
,
τ
)
is the solution of
(74)
K
˙
1
=
-
A
1
′
(
t
)
K
1
-
K
1
A
1
(
t
)
-
Q
1
(
t
)
+
K
1
S
1
(
t
)
K
1
,
K
˙
1
=
-
A
1
′
(
t
)
K
1
-
K
1
A
1
(
t
)
-
Q
1
K
1
(
τ
)
=
k
(
τ
,
τ
)
;
and
K
~
(
t
,
τ
)
is the solution of
(75)
K
˙
2
=
-
A
2
′
(
t
)
K
2
-
K
2
A
2
(
t
)
-
Q
2
(
t
)
+
K
2
S
2
(
t
)
K
2
,
K
˙
2
=
-
A
2
′
(
t
)
K
2
-
K
2
A
2
(
t
)
-
Q
2
K
2
(
τ
)
=
K
~
(
τ
,
τ
)
.
Therefore, we have
(76)
(
Q
2
A
2
′
A
2
-
S
2
)
(
t
)
-
(
Q
1
A
1
′
A
1
-
S
1
)
(
t
)
=
(
1
-
γ
2
γ
K
2
σ
r
2
0
0
0
)
≥
0
2
×
2
,
and
K
2
(
τ
)
≥
K
1
(
τ
)
. Applying Proposition 9,
k
(
t
,
τ
)
≤
K
~
(
t
,
τ
)
for all
t
∈
[
0
,
τ
]
. It is because
σ
r
2
/
2
≥
0
and
-
(
(
1
-
γ
)
/
2
γ
)
K
2
σ
r
2
≤
0
in ODE (72), and
k
(
τ
,
τ
)
=
K
(
τ
,
T
)
≥
0
, which is proved in Lemma 10, that
k
(
t
,
τ
)
exists for all
t
∈
[
0
,
τ
]
by Theorem 4.1 in [16].
Theorem 14.
The function,
(77)
v
~
(
t
,
Y
,
r
)
=
Y
1
-
γ
ψ
μ
*
(
t
,
τ
)
(
ψ
z
(
1
-
γ
)
-
1
)
1
-
γ
e
K
~
(
t
,
τ
)
r
+
M
~
(
t
,
τ
)
,
is bounded for all
0
≤
t
≤
τ
≤
T
if
K
(
t
,
T
)
has no finite escape time on
[
0
,
T
]
.
Proof.
Suppose that
K
(
t
,
T
)
has no finite escape time on
[
0
,
T
]
. It implies that
K
~
(
t
,
τ
)
are bounded for
0
≤
t
≤
τ
≤
T
, and so is
M
~
(
t
,
τ
)
. Since the moment generating function of
μ
*
(
t
,
τ
)
and
z
,
ψ
μ
*
(
t
,
τ
)
(
·
)
and
ψ
z
(
·
)
, then
v
~
(
t
,
Y
,
β
)
is bounded.
The above theorem shows the uniform integrability of
{
V
(
τ
,
Y
u
*
(
τ
)
,
β
(
τ
)
}
τ
. The calculation of
(78)
E
[
ln
Y
(
τ
)
+
∫
τ
T
∥
Λ
(
s
)
∥
2
2
d
s
-
E
[
z
]
∫
τ
T
E
[
μ
(
s
)
]
d
s
+
∫
τ
T
(
e
-
∫
τ
s
b
(
η
)
d
η
r
(
τ
)
+
∫
τ
s
e
-
∫
η
s
b
(
θ
)
d
θ
a
(
η
)
d
η
)
×
(
1
-
λ
r
(
s
)
2
2
)
d
s
∣
G
t
]
will be shown as follows:
(79)
E
[
(
1
-
λ
r
(
s
)
2
2
)
d
s
∣
G
t
]
ln
Y
(
τ
)
+
∫
τ
T
∥
Λ
(
s
)
∥
2
2
d
s
-
E
[
z
]
∫
τ
T
E
[
μ
(
s
)
]
d
s
+
∫
τ
T
(
e
-
∫
τ
s
b
(
η
)
d
η
r
(
τ
)
+
∫
τ
s
e
-
∫
η
s
b
(
θ
)
d
θ
a
(
η
)
d
η
)
×
(
1
-
λ
r
(
s
)
2
2
)
d
s
∣
G
t
]
=
∫
τ
T
∥
Λ
(
s
)
∥
2
2
d
s
-
E
[
z
]
∫
τ
T
E
[
μ
(
s
)
]
d
s
+
E
[
(
1
-
λ
r
(
s
)
2
2
)
d
s
∣
G
t
]
ln
Y
u
*
(
τ
)
+
∫
τ
T
(
e
-
∫
τ
s
b
(
η
)
d
η
r
(
τ
)
+
∫
τ
s
e
-
∫
η
s
b
(
θ
)
d
θ
a
(
η
)
d
η
)
×
(
1
-
λ
r
(
s
)
2
2
)
d
s
∣
G
t
]
=
∫
τ
T
∥
Λ
(
s
)
∥
2
2
d
s
-
E
[
z
]
∫
τ
T
E
[
μ
(
s
)
]
d
s
+
E
[
+
∫
τ
T
E
[
r
(
s
)
∣
G
τ
]
(
1
-
λ
r
(
s
)
2
2
)
d
s
∣
G
t
]
ln
Y
u
*
(
τ
)
+
∫
τ
T
E
[
r
(
s
)
∣
G
τ
]
(
1
-
λ
r
(
s
)
2
2
)
d
s
∣
G
t
]
=
∫
τ
T
∥
Λ
(
s
)
∥
2
2
d
s
-
E
[
z
]
∫
τ
T
E
[
μ
(
s
)
]
d
s
+
E
[
ln
Y
u
*
(
τ
)
∣
G
t
]
+
E
[
∫
τ
T
E
[
r
(
s
)
∣
G
τ
]
(
1
-
λ
r
(
s
)
2
2
)
d
s
∣
G
t
]
=
∫
τ
T
∥
Λ
(
s
)
∥
2
2
d
s
-
E
[
z
]
∫
τ
T
E
[
μ
(
s
)
]
d
s
+
ln
Y
(
t
)
+
∫
t
τ
∥
Λ
(
s
)
∥
2
2
d
s
-
E
[
z
]
∫
t
τ
E
[
μ
(
s
)
]
d
s
+
∫
t
τ
E
[
r
(
s
)
∣
G
t
]
(
1
-
λ
r
(
s
)
2
2
)
d
s
+
E
[
∫
τ
T
E
[
r
(
s
)
∣
G
τ
]
(
1
-
λ
r
(
s
)
2
2
)
d
s
∣
G
t
]
=
∫
t
T
∥
Λ
(
s
)
∥
2
2
d
s
-
E
[
z
]
∫
t
T
E
[
μ
(
s
)
]
d
s
+
ln
Y
(
t
)
+
E
[
E
[
∫
t
T
r
(
s
)
(
1
-
λ
r
(
s
)
2
2
)
d
s
∣
G
τ
]
∣
G
t
]
=
∫
t
T
∥
Λ
(
s
)
∥
2
2
d
s
-
E
[
z
]
∫
t
T
E
[
μ
(
s
)
]
d
s
+
ln
Y
(
t
)
+
∫
t
T
E
[
r
(
s
)
∣
G
t
]
(
1
-
λ
r
(
s
)
2
2
)
d
s
=
∫
t
T
∥
Λ
(
s
)
∥
2
2
d
s
-
E
[
z
]
∫
t
T
E
[
μ
(
s
)
]
d
s
+
ln
Y
(
t
)
+
∫
t
T
(
e
-
∫
t
s
b
(
η
)
d
η
r
(
t
)
+
∫
t
s
e
-
∫
η
s
b
(
θ
)
d
θ
a
(
η
)
d
η
)
×
(
1
-
λ
r
(
s
)
2
2
)
d
s
<
+
∞
.
Hence,
{
J
(
τ
,
Y
u
*
(
τ
)
,
β
(
τ
)
}
τ
is uniformly integrable.