# Viscosity Solution of Mean-Variance Portfolio Selection of a Jump Markov Process with No-Shorting Constraints

Authors: Moussa Kounta
Journal: Journal of Applied Mathematics
 (2016)

---

## Abstract

We consider the so-called mean-variance portfolio selection problem in continuous time under the constraint that the short-selling of stocks is prohibited where all the market coefficients are random processes. In this situation the Hamilton-Jacobi-Bellman (HJB) equation of the value function of the auxiliary problem becomes a coupled system of backward stochastic partial differential equation. In fact, the value functionV often does not have the smoothness properties needed to interpret it as a solution to the dynamic programming partial differential equation in the usual (classical) sense; however, in such cases V can be interpreted as a viscosity solution. Here we show the unicity of the viscosity solution and we see that the optimal and the value functions are piecewise linear functions based on some Riccati differential equations. In particular we solve the open problem posed by Li and Zhou and Zhou and Yin.


---

## Body

## 1. Introduction

The mean-variance approach proposed in 1952 by the Nobel prize winning economist Markowitz [1] has become the foundation of modern finance by discovering the static mean-variance portfolio selection formulation in a market in which shorting is not allowed. This theory has inspired numerous extensions and applications. For instance, Li and Ng [2] and Zhou and Li [3] successfully extended the unconstrained mean-variance portfolio selection formulation to the multiperiod setting. Zhou and Yin [4] consider the mean-variance portfolio selection problem in continuous time where the market parameters including the bank interest rate and the appreciation and volatility rates of the stocks depend on the market mode that switches among a finite number of states where random regime switching is assumed to be independent of the underlying Brownian motion. This essentially renders the underlying market incomplete. A Markov chain modulated diffusion formulation is employed to model the problem and Zhou and Yin [4] use the techniques of stochastic linear quadratic (LQ) control to derive mean-variance efficient portfolios and efficient frontier based on solutions of two systems of linear ordinary differential equations.After Li and Ng published [2], Markowitz suggested that one of them extends the results to the dynamic mean-variance formulation with no-shorting constraint and proposed a conjecture of a piecewise quadratic value function for such a situation. Influenced by Markowitz’s comments, Li et al. [5] formulated the LQ control problem by constraining the control portfolio to take nonnegative values due to the no-shorting restriction on the market mode (not random processes). They derived the optimal portfolio policy for the continuous-time mean-variance model with no-shorting constraint using the duality method [6].However, there are several interesting problems that deserve further investigation; for instance, Li et al. [5] open a problem by stating in their conclusion that “an immediate open problem is to extend the results in this paper to the case where all the market coefficients are random processes.” In this paper we solve this problem.By making use of the techniques of LQ control, we see that, in an attempt to pursue the method of dynamic programming in the auxiliary problem, the value function which is a generalized solution to the Hamilton-Jacobi equation coupled is not smooth enough to satisfy the dynamic programming equations in the classical or usual sense. A difficulty with the concept of generalized solution is that the dynamic programming together with the boundary data typically has many generalized solutions. Among them, there is one provided by Crandall and Lions [7], called the viscosity solution, which is the natural generalized solution. This unique viscosity solution turns out to coincide with the value function V [8]. The central component of our solution to the problem of Li et al. [5] is the proof of the unicity of the viscosity solution of the value function of the auxiliary problem, which we establish by adapting the techniques of [9]. By making use of the duality method, we also derive a solution for efficient portfolio. The value function of the auxiliary problem depends on a set of Riccati differential equations and we use the Magnus approach to provide the solution. A work is in progress to develop numerical implementation. This will be subject of a future publication.


## 2. Viscosity Solutions for Weakly Coupled Systems of Second-Order Hamilton-Jacobi-Bellman Equation

### 2.1. Notation

We make use of the following notations:(i)
(
Ω
,
F
,
P
): a fixed probability space on which we defined standard n-dimensional Brownian motion W
(
t
)
≡
W
1
t
,
…
,
W
n
t
′ and continuous-time stationary Markov chain α
(
t
) taking value in a finite state space M
=
{
1,2
,
…
,
m
} such that W
(
t
) and α
(
t
) are independent of each other. The Markov chain has a generator Q
=
(
q
i
j
)
m
×
m and stationary transition probabilities: (1)
p
i
j
t
=
p
α
t
=
j
∣
α
0
=
i
,
t
≥
0
,
i
,
j
=
1,2
,
…
,
m
.
(ii)
DefineF
t
=
σ
W
(
s
)
,
α
(
s
)
:
0
≤
s
≤
t.
(iii)
B
(
Σ
)
=
σ
-
a
l
g
e
b
r
a
o
f
B
o
r
e
l
s
e
t
s
o
f
Σ.
(iv)
Consider the following:(2)
E
t
x
Φ
x
s
=
∫
Σ
Φ
y
P
^
t
,
x
,
s
,
d
y
,
t
<
s
.
P
^
t
,
x
,
s
,
B
=
P
x
s
∈
B
∣
x
t
=
x
,
∀
B
∈
B
Σ
.
(v)
Hilbert spaceH with the norm ·
H: define the Banach space (3)
L
F
2
0
,
T
;
H
=
φ
·
∣
φ
·
is  an
F
t
-adapted
,
H
-valued  measurable  process  on
a
,
b
,
E
∫
a
b
φ
t
,
ω
H
d
t
<
∞
,
with norm(4)
φ
·
F
,
2
=
E
∫
a
b
φ
t
,
ω
H
d
t
<
∞
.
(vi)
M
′: the transpose of any vector or matrix.
(vii)
M
j: the jth component of any vector M;
we will use indifferently this notationM
(
t
,
x
,
u
,
i
)
≡
M
i
(
t
,
x
,
u
).
(viii)
C
(
[
0
,
T
]
;
X
): the Banach space of X-valued continuous functions on [
0
,
T
].
(ix)
C
2
(
[
0
,
T
]
×
R
n
): the space of all twice continuously differentiable functions on [
0
,
T
]
×
R
n.
(x)
ConsiderD
x
=
∂
(
·
)
/
∂
x, D
x
2
=
∂
2
(
·
)
/
∂
x
2
,
D
α
(
·
)
=
∂
|
α
|
(
·
)
/
∂
x
1
α
1
⋯
∂
x
n
α
n, and f
˙
(
t
)
=
d
/
d
t
f
(
t
), ∂
t
=
∂
/
∂
t.
(xi)
Consider(5)
Q
=
0
,
T
×
D
,
D
⊂
R
.
(xii)
Kronecker delta symbol:(6)
δ
i
j
t
=
0
if
i
≠
j
1
if
i
=
j
.
(xiii)
[
A
,
B
]
≡
A
B
-
B
A (Lie bracket), A
,
B matrices with appropriate dimension.
(xiv)
ConsiderW
k
,
p
(
Ω
)
=
{
u
∈
L
p
Ω
:
D
α
u
∈
L
p
Ω
,
1
≤
p
≤
∞
,
∀
|
α
|
≤
k
}
.
(xv)
C
b
(
Σ
) is bounded function in Σ.
(xvi)
IfF is a real-valued function on a set U which has a minimum on U, then (7)
argmin
v
∈
U
⁡
F
v
=
v
∗
∈
U
:
F
v
∗
≤
F
v
,
∀
v
∈
U
.


## 2.1. Notation

We make use of the following notations:(i)
(
Ω
,
F
,
P
): a fixed probability space on which we defined standard n-dimensional Brownian motion W
(
t
)
≡
W
1
t
,
…
,
W
n
t
′ and continuous-time stationary Markov chain α
(
t
) taking value in a finite state space M
=
{
1,2
,
…
,
m
} such that W
(
t
) and α
(
t
) are independent of each other. The Markov chain has a generator Q
=
(
q
i
j
)
m
×
m and stationary transition probabilities: (1)
p
i
j
t
=
p
α
t
=
j
∣
α
0
=
i
,
t
≥
0
,
i
,
j
=
1,2
,
…
,
m
.
(ii)
DefineF
t
=
σ
W
(
s
)
,
α
(
s
)
:
0
≤
s
≤
t.
(iii)
B
(
Σ
)
=
σ
-
a
l
g
e
b
r
a
o
f
B
o
r
e
l
s
e
t
s
o
f
Σ.
(iv)
Consider the following:(2)
E
t
x
Φ
x
s
=
∫
Σ
Φ
y
P
^
t
,
x
,
s
,
d
y
,
t
<
s
.
P
^
t
,
x
,
s
,
B
=
P
x
s
∈
B
∣
x
t
=
x
,
∀
B
∈
B
Σ
.
(v)
Hilbert spaceH with the norm ·
H: define the Banach space (3)
L
F
2
0
,
T
;
H
=
φ
·
∣
φ
·
is  an
F
t
-adapted
,
H
-valued  measurable  process  on
a
,
b
,
E
∫
a
b
φ
t
,
ω
H
d
t
<
∞
,
with norm(4)
φ
·
F
,
2
=
E
∫
a
b
φ
t
,
ω
H
d
t
<
∞
.
(vi)
M
′: the transpose of any vector or matrix.
(vii)
M
j: the jth component of any vector M;
we will use indifferently this notationM
(
t
,
x
,
u
,
i
)
≡
M
i
(
t
,
x
,
u
).
(viii)
C
(
[
0
,
T
]
;
X
): the Banach space of X-valued continuous functions on [
0
,
T
].
(ix)
C
2
(
[
0
,
T
]
×
R
n
): the space of all twice continuously differentiable functions on [
0
,
T
]
×
R
n.
(x)
ConsiderD
x
=
∂
(
·
)
/
∂
x, D
x
2
=
∂
2
(
·
)
/
∂
x
2
,
D
α
(
·
)
=
∂
|
α
|
(
·
)
/
∂
x
1
α
1
⋯
∂
x
n
α
n, and f
˙
(
t
)
=
d
/
d
t
f
(
t
), ∂
t
=
∂
/
∂
t.
(xi)
Consider(5)
Q
=
0
,
T
×
D
,
D
⊂
R
.
(xii)
Kronecker delta symbol:(6)
δ
i
j
t
=
0
if
i
≠
j
1
if
i
=
j
.
(xiii)
[
A
,
B
]
≡
A
B
-
B
A (Lie bracket), A
,
B matrices with appropriate dimension.
(xiv)
ConsiderW
k
,
p
(
Ω
)
=
{
u
∈
L
p
Ω
:
D
α
u
∈
L
p
Ω
,
1
≤
p
≤
∞
,
∀
|
α
|
≤
k
}
.
(xv)
C
b
(
Σ
) is bounded function in Σ.
(xvi)
IfF is a real-valued function on a set U which has a minimum on U, then (7)
argmin
v
∈
U
⁡
F
v
=
v
∗
∈
U
:
F
v
∗
≤
F
v
,
∀
v
∈
U
.


## 3. Notion of Viscosity Solution

We consider the following coupled system of backward PDEs:(8)
∂
t
V
i
t
,
x
+
inf
u
≥
0
⁡
1
2
g
¯
i
t
,
x
t
,
u
g
¯
i
t
,
x
t
,
u
′
D
x
2
V
i
t
,
x
+
f
¯
i
t
,
x
t
,
u
D
x
V
i
t
,
x
-
∑
j
≠
i
q
i
j
V
i
t
,
x
-
V
j
t
,
x
=
0
,
V
i
T
,
x
=
g
x
t
,
x
∈
Q
w
h
e
r
e
u
t
=
u
1
t
,
…
,
u
m
t
′
,and the conditions on matrix q
i
j
1
≤
i
,
j
≤
n are(9)
q
k
l
>
0
,
for
k
≠
l
,
q
k
k
<
0
,
∑
l
=
1
m
q
k
l
=
0
f
o
r
k
=
1
,
…
,
m
.We suppose(10)
f
¯
i
t
,
x
t
;
u
,
g
¯
i
t
,
x
t
,
u
∈
W
2
,
∞
Q
.Under appropriate regularity assumptions on ∂
Q and the coefficients, we define and prove existence and uniqueness results of the viscosity solutions to (8).


### 3.1. Viscosity Solution Definition

It is well known that (8) does not in general have classical smooth solutions. We define a generalized concept of solution called a viscosity solution [7].Definition 1 (w
∈
(
C
b
2
(
Q
)
)
m).
w is a viscosity subsolution (supersolution) of system (8), if, for all ϕ
i
∈
C
2
(
Q
),(11)
∂
t
ϕ
i
t
0
,
x
0
+
inf
u
≥
0
⁡
1
2
g
¯
i
t
0
,
x
0
,
u
g
¯
i
t
0
,
x
0
,
u
′
D
x
2
ϕ
i
t
0
,
x
0
+
f
¯
i
t
0
,
x
0
,
u
D
x
ϕ
i
t
0
,
x
0
-
∑
j
≠
i
q
i
j
w
i
t
0
,
x
0
-
w
j
t
0
,
x
0
≥
0
,
(12)
∂
t
ϕ
i
t
0
,
x
0
+
inf
u
≥
0
⁡
1
2
g
¯
i
t
0
,
x
0
,
u
g
¯
i
t
0
,
x
0
,
u
′
D
x
2
ϕ
i
t
0
,
x
0
+
f
¯
i
t
0
,
x
0
,
u
D
x
ϕ
i
t
0
,
x
0
-
∑
j
≠
i
q
i
j
w
i
t
0
,
x
0
-
w
j
t
0
,
x
0
≤
0
,respectively, whenever w
i
-
ϕ
i has a local maximum (minimum) at (
t
0
,
x
0
)
∈
Q; w is a viscosity solution if it is both a viscosity subsolution and supersolution.


### 3.2. Uniqueness Result

Next, we can let(i)
(13)
H
i
t
,
x
,
p
,
A
=
inf
u
≥
0
⁡
1
2
g
¯
i
t
,
x
,
u
g
¯
i
t
,
x
,
u
′
A
+
f
¯
i
t
,
x
,
u
p
=
-
sup
u
≥
0
⁡
-
1
2
g
¯
i
t
,
x
,
u
g
¯
i
t
,
x
,
u
′
A
-
f
¯
i
t
,
x
,
u
p
,and we assume(ii)
(a)
|
(
f
¯
i
)
t
|
+
|
(
f
¯
i
)
x
|
≤
C
,
|
(
g
¯
i
)
t
|
+
|
(
g
¯
i
)
x
|
≤
C;
(b)
f
¯
i
t
,
x
,
u
≤
C
1
+
x
+
u;
(c)
|
g
¯
i
(
t
,
x
,
u
)
|
≤
C
(
1
+
|
x
|
+
|
u
|
).Lemma 2 (see [8], let H
i be as in (13)).
Assume ((ii)(a)–(c)).
Then, there exists a continuous functionω
:
[
0
,
∞
)
→
[
0
,
∞
) that satisfies ω
(
0
)
=
0 such that(14)
H
i
t
,
y
,
β
x
-
y
,
B
-
H
i
t
,
x
,
β
x
-
y
,
A
≤
ω
β
x
-
y
2
+
x
-
y
,for every (
t
,
x
)
,
(
t
,
y
)
∈
Q
,
β
>
0, and symmetric matrices A, B satisfying (15)
-
3
β
I
0
0
I
≤
A
0
0
-
B
≤
3
β
I
-
I
-
I
I
,where I is the identity matrix with appropriate dimension.Proposition 3.
Suppose assumptions (9) and (10) hold and w
k is a viscosity subsolution of (8) and v
k is a supersolution of (8). Ifw
k
(
t
,
x
)
≤
v
k
(
t
,
x
) on ∂
Q
,
k
=
1
,
…
,
m, then
w
k
(
t
,
x
)
≤
v
k
(
t
,
x
) on Q
,
k
=
1
,
…
,
m.Proof.
Suppose that there does not exist an index,s and (
l
,
z
)
∈
Q, such that(16)
w
s
-
v
s
l
,
z
=
max
x
,
t
,
k
⁡
w
k
-
v
k
t
,
x
>
0
.
(i)
If(
l
,
z
)
∈
∂
Q, we are done.
(ii)
Assume(
l
,
z
)
∈
Q; let(17)
ϕ
r
ε
t
,
x
,
y
=
w
k
t
,
x
-
v
k
t
,
y
-
1
ε
2
x
-
y
2
-
ε
t
-
l
2
.There exists an index r and (
t
0
,
x
0
,
y
0
)
=
(
t
0
ε
,
x
0
ε
,
y
0
ε
) such that (18)
ϕ
r
ε
t
0
,
x
0
,
y
0
=
max
x
,
t
,
y
,
k
⁡
ϕ
k
ε
t
,
x
,
y
.We now show(19)
w
r
t
0
,
x
0
-
v
r
t
0
,
y
0
≥
0
.But (20)
ϕ
s
ε
l
,
z
,
z
=
w
s
l
,
z
-
v
s
l
,
z
≤
ϕ
r
ε
t
0
,
x
0
,
y
0implies (21)
1
ε
2
x
0
-
y
0
2
+
ε
t
0
-
l
2
≤
w
r
t
0
,
x
0
-
v
r
t
0
,
y
0
-
w
s
l
,
z
-
v
s
l
,
z
.Since w
s
(
l
,
z
)
-
v
s
(
l
,
z
)
>
0, (22)
0
≤
1
ε
2
x
0
-
y
0
2
+
ε
t
0
-
l
2
≤
w
r
t
0
,
x
0
-
v
r
t
0
,
y
0
-
w
s
l
,
z
-
v
s
l
,
z
≤
w
r
t
0
,
x
0
-
v
r
t
0
,
y
0
.
(i) Sincew is a viscosity subsolution of (8) and the function(23)
Ψ
r
,
1
ε
:
x
,
t
⟼
ϕ
r
ε
t
,
x
,
y
0has a maximum at (
t
0
,
x
0
), set (24)
φ
r
,
1
ε
t
,
x
=
v
r
t
,
y
0
+
1
ε
2
x
-
y
0
2
+
ε
t
-
l
2
,then w
r
(
t
,
x
)
-
φ
r
,
1
ε
(
t
,
x
), has a maximum at (
t
0
,
x
0
), and hence(25)
∂
w
r
-
φ
r
,
1
ε
∂
t
t
0
,
x
0
=
∂
w
r
-
φ
r
,
1
ε
∂
x
t
0
,
x
0
=
0
⟹
∂
φ
r
,
1
ε
∂
t
t
0
,
x
0
=
∂
w
r
∂
t
t
0
,
x
0
,
∂
φ
r
,
1
ε
∂
x
t
0
,
x
0
=
∂
w
r
∂
x
t
0
,
x
0
,
∂
Ψ
r
,
1
ε
∂
x
t
0
,
x
0
=
0
⟺
∂
w
r
∂
x
t
0
,
x
0
-
2
ε
2
x
0
-
y
0
=
0
,
∂
Ψ
r
,
1
ε
∂
t
t
0
,
x
0
=
0
⟺
∂
w
r
∂
t
t
0
,
x
0
-
∂
v
r
t
0
,
x
0
∂
t
t
0
,
x
0
-
2
ε
t
0
-
l
=
0
,
(26)
0
≤
∂
φ
r
,
1
ε
t
0
,
x
0
∂
t
-
H
r
t
0
,
x
0
,
2
ε
2
x
0
-
y
0
,
D
x
2
φ
r
,
1
ε
t
0
,
x
0
-
∑
k
≠
r
q
r
k
w
r
t
0
,
x
0
-
w
k
t
0
,
x
0
.
(ii) Sincev is a viscosity supersolution of (8) and the function(27)
Ψ
r
,
2
ε
:
x
,
t
⟼
-
ϕ
r
ε
t
,
x
0
,
yhas a minimum at (
t
0
,
y
0
), set (28)
φ
r
,
2
ε
t
,
y
=
w
r
t
,
x
0
-
1
ε
2
x
0
-
y
2
-
ε
t
-
l
2then φ
r
,
2
ε
(
t
,
x
)
-
v
r
(
t
,
x
), has a maximum at (
t
0
,
x
0
), and hence(29)
∂
φ
r
,
2
ε
-
v
r
∂
t
t
0
,
y
0
=
∂
φ
r
,
2
ε
-
v
r
∂
x
t
0
,
y
0
=
0
⟹
∂
φ
r
,
2
ε
∂
t
t
0
,
y
0
=
∂
v
r
∂
t
t
0
,
y
0
,
∂
φ
r
,
2
ε
∂
x
t
0
,
y
0
=
∂
v
r
∂
x
t
0
,
y
0
,
∂
Ψ
r
,
2
ε
∂
y
t
0
,
y
0
=
0
⟺
+
∂
v
r
∂
y
t
0
,
y
0
-
2
ε
2
x
0
-
y
0
=
0
,
∂
Ψ
r
,
2
ε
∂
t
t
0
,
x
0
=
0
⟺
-
∂
w
r
∂
t
t
0
,
x
0
+
∂
v
r
t
0
,
x
0
∂
t
t
0
,
x
0
+
2
ε
t
0
-
l
=
0
,
(30)
∂
φ
r
,
2
ε
t
0
,
y
0
∂
t
-
H
r
t
0
,
x
0
,
2
ε
2
x
0
-
y
0
,
D
y
2
φ
r
,
2
ε
t
0
,
y
0
-
∑
k
≠
r
q
r
k
v
r
t
0
,
y
0
-
v
k
t
0
,
y
0
≤
0
.By combining (26) and (30),(31)
0
≤
∂
w
r
t
0
,
x
0
∂
t
-
∂
v
r
t
0
,
y
0
∂
t
+
H
r
t
0
,
y
0
,
2
ε
2
x
0
-
y
0
,
D
y
2
φ
r
,
2
ε
t
0
,
y
0
-
H
r
t
0
,
x
0
,
2
ε
2
x
0
-
y
0
,
D
x
2
φ
r
,
1
ε
t
0
,
x
0
-
∑
k
≠
r
q
r
k
w
r
t
0
,
x
0
-
w
k
t
0
,
x
0
+
∑
k
≠
r
q
r
k
v
r
t
0
,
y
0
-
v
k
t
0
,
y
0
,and by Lemma 2
(32)
0
≤
2
ε
t
0
-
l
+
ω
2
ε
2
x
0
-
y
0
2
+
x
0
-
y
0
-
∑
k
≠
r
q
r
k
w
r
t
0
,
x
0
-
w
k
t
0
,
x
0
+
∑
k
≠
r
q
r
k
v
r
t
0
,
y
0
-
v
k
t
0
,
y
0
,we obtain (33)
0
≤
2
ε
t
0
-
l
+
ω
2
ε
2
x
0
-
y
0
2
+
x
0
-
y
0
-
∑
k
≠
r
q
r
k
w
r
t
0
,
x
0
-
v
r
t
0
,
y
0
+
∑
k
≠
r
q
r
k
w
k
t
0
,
x
0
-
v
k
t
0
,
y
0
.We have -
q
r
k
<
0 and w
r
(
t
0
,
x
0
)
-
v
r
(
t
0
,
y
0
)
>
0 (19) and hence (34)
0
≤
2
ε
t
0
-
l
+
ω
2
ε
2
x
0
-
y
0
2
+
x
0
-
y
0
+
∑
k
≠
r
q
r
k
w
k
t
0
,
x
0
-
v
k
t
0
,
y
0
.Since w
k
(
t
0
,
x
0
)
-
v
k
(
t
0
,
y
0
)
≤
w
r
(
t
0
,
x
0
)
-
v
r
(
t
0
,
y
0
) and ∑
k
≠
r
‍
q
r
k
=
-
q
k
k, we obtain (35)
0
≤
2
ε
t
0
-
l
+
ω
2
ε
2
x
0
-
y
0
2
+
x
0
-
y
0
-
q
k
k
w
r
t
0
,
x
0
-
v
r
t
0
,
y
0
.Thus, (36)
w
s
l
,
z
-
v
s
l
,
z
≤
ϕ
r
ε
t
0
,
x
0
,
y
0
≤
w
r
t
0
,
x
0
-
v
r
t
0
,
y
0
≤
1
q
k
k
2
ε
t
0
-
l
+
1
q
k
k
ω
2
ε
2
x
0
-
y
0
2
+
x
0
-
y
0
.To finish the proof, we need to show (37)
ω
2
ε
2
x
0
-
y
0
2
+
x
0
-
y
0
⟶
0
,
a
s
ε
⟶
0
.Let (38)
h
q
=
Sup
⁡
v
r
t
,
x
-
v
r
t
,
y
:
t
∈
0
,
T
,
x
,
y
∈
Q
¯
x
-
y
2
≤
q
,so that for any (
t
,
x
) and (
t
,
y
)
∈
Q
¯
(39)
v
r
t
,
x
-
v
r
t
,
y
≤
h
x
-
y
2
.Since (
x
0
,
t
0
,
y
0
,
r
) maximizes ϕ
r
ε over Q, (40)
w
r
t
0
,
x
0
-
v
r
t
0
,
y
0
-
1
ε
2
x
0
-
y
0
2
-
ε
t
0
-
l
2
≥
w
r
t
0
,
x
0
-
v
r
t
0
,
x
0
-
1
ε
2
x
0
-
x
0
2
-
ε
t
0
-
l
2
.We obtain (41)
1
ε
2
x
0
-
y
0
2
≤
v
r
t
0
,
x
0
-
v
r
t
0
,
y
0
≤
h
x
0
-
y
0
2
.Since h is bounded by some constant K, this implies that (42)
2
ε
2
x
0
-
y
0
2
≤
K
.The definition of h yields (43)
2
ε
2
x
0
-
y
0
2
+
x
0
-
y
0
≤
h
K
ε
2
2
+
ε
K
2
,and we obtain (44)
ω
2
ε
2
x
0
-
y
0
2
+
x
0
-
y
0
≤
ω
h
K
ε
2
2
+
ε
K
2
,and we obtain (45)
w
s
l
,
z
-
v
s
l
,
z
≤
0
a
s
ε
⟶
0
,which is a contradiction to (16).Corollary 4.
The viscosity solution satisfying the boundary condition is unique.Proof.
Ifv
k
1 and v
k
2 are 2 viscosity solutions such that v
k
1
=
v
k
2 on ∂
Q, then(i)
v
k
1
≤
v
k
2 on ∂
Q
⇒
v
k
1
≤
v
k
2 on Q accordingly (Proposition 3);
(ii)
v
k
1
≥
v
k
2 on ∂
Q
⇒
v
k
1
≥
v
k
2 on Q accordingly (Proposition 3).Hence v
k
1
=
v
k
2 on Q.


## 3.1. Viscosity Solution Definition

It is well known that (8) does not in general have classical smooth solutions. We define a generalized concept of solution called a viscosity solution [7].Definition 1 (w
∈
(
C
b
2
(
Q
)
)
m).
w is a viscosity subsolution (supersolution) of system (8), if, for all ϕ
i
∈
C
2
(
Q
),(11)
∂
t
ϕ
i
t
0
,
x
0
+
inf
u
≥
0
⁡
1
2
g
¯
i
t
0
,
x
0
,
u
g
¯
i
t
0
,
x
0
,
u
′
D
x
2
ϕ
i
t
0
,
x
0
+
f
¯
i
t
0
,
x
0
,
u
D
x
ϕ
i
t
0
,
x
0
-
∑
j
≠
i
q
i
j
w
i
t
0
,
x
0
-
w
j
t
0
,
x
0
≥
0
,
(12)
∂
t
ϕ
i
t
0
,
x
0
+
inf
u
≥
0
⁡
1
2
g
¯
i
t
0
,
x
0
,
u
g
¯
i
t
0
,
x
0
,
u
′
D
x
2
ϕ
i
t
0
,
x
0
+
f
¯
i
t
0
,
x
0
,
u
D
x
ϕ
i
t
0
,
x
0
-
∑
j
≠
i
q
i
j
w
i
t
0
,
x
0
-
w
j
t
0
,
x
0
≤
0
,respectively, whenever w
i
-
ϕ
i has a local maximum (minimum) at (
t
0
,
x
0
)
∈
Q; w is a viscosity solution if it is both a viscosity subsolution and supersolution.


## 3.2. Uniqueness Result

Next, we can let(i)
(13)
H
i
t
,
x
,
p
,
A
=
inf
u
≥
0
⁡
1
2
g
¯
i
t
,
x
,
u
g
¯
i
t
,
x
,
u
′
A
+
f
¯
i
t
,
x
,
u
p
=
-
sup
u
≥
0
⁡
-
1
2
g
¯
i
t
,
x
,
u
g
¯
i
t
,
x
,
u
′
A
-
f
¯
i
t
,
x
,
u
p
,and we assume(ii)
(a)
|
(
f
¯
i
)
t
|
+
|
(
f
¯
i
)
x
|
≤
C
,
|
(
g
¯
i
)
t
|
+
|
(
g
¯
i
)
x
|
≤
C;
(b)
f
¯
i
t
,
x
,
u
≤
C
1
+
x
+
u;
(c)
|
g
¯
i
(
t
,
x
,
u
)
|
≤
C
(
1
+
|
x
|
+
|
u
|
).Lemma 2 (see [8], let H
i be as in (13)).
Assume ((ii)(a)–(c)).
Then, there exists a continuous functionω
:
[
0
,
∞
)
→
[
0
,
∞
) that satisfies ω
(
0
)
=
0 such that(14)
H
i
t
,
y
,
β
x
-
y
,
B
-
H
i
t
,
x
,
β
x
-
y
,
A
≤
ω
β
x
-
y
2
+
x
-
y
,for every (
t
,
x
)
,
(
t
,
y
)
∈
Q
,
β
>
0, and symmetric matrices A, B satisfying (15)
-
3
β
I
0
0
I
≤
A
0
0
-
B
≤
3
β
I
-
I
-
I
I
,where I is the identity matrix with appropriate dimension.Proposition 3.
Suppose assumptions (9) and (10) hold and w
k is a viscosity subsolution of (8) and v
k is a supersolution of (8). Ifw
k
(
t
,
x
)
≤
v
k
(
t
,
x
) on ∂
Q
,
k
=
1
,
…
,
m, then
w
k
(
t
,
x
)
≤
v
k
(
t
,
x
) on Q
,
k
=
1
,
…
,
m.Proof.
Suppose that there does not exist an index,s and (
l
,
z
)
∈
Q, such that(16)
w
s
-
v
s
l
,
z
=
max
x
,
t
,
k
⁡
w
k
-
v
k
t
,
x
>
0
.
(i)
If(
l
,
z
)
∈
∂
Q, we are done.
(ii)
Assume(
l
,
z
)
∈
Q; let(17)
ϕ
r
ε
t
,
x
,
y
=
w
k
t
,
x
-
v
k
t
,
y
-
1
ε
2
x
-
y
2
-
ε
t
-
l
2
.There exists an index r and (
t
0
,
x
0
,
y
0
)
=
(
t
0
ε
,
x
0
ε
,
y
0
ε
) such that (18)
ϕ
r
ε
t
0
,
x
0
,
y
0
=
max
x
,
t
,
y
,
k
⁡
ϕ
k
ε
t
,
x
,
y
.We now show(19)
w
r
t
0
,
x
0
-
v
r
t
0
,
y
0
≥
0
.But (20)
ϕ
s
ε
l
,
z
,
z
=
w
s
l
,
z
-
v
s
l
,
z
≤
ϕ
r
ε
t
0
,
x
0
,
y
0implies (21)
1
ε
2
x
0
-
y
0
2
+
ε
t
0
-
l
2
≤
w
r
t
0
,
x
0
-
v
r
t
0
,
y
0
-
w
s
l
,
z
-
v
s
l
,
z
.Since w
s
(
l
,
z
)
-
v
s
(
l
,
z
)
>
0, (22)
0
≤
1
ε
2
x
0
-
y
0
2
+
ε
t
0
-
l
2
≤
w
r
t
0
,
x
0
-
v
r
t
0
,
y
0
-
w
s
l
,
z
-
v
s
l
,
z
≤
w
r
t
0
,
x
0
-
v
r
t
0
,
y
0
.
(i) Sincew is a viscosity subsolution of (8) and the function(23)
Ψ
r
,
1
ε
:
x
,
t
⟼
ϕ
r
ε
t
,
x
,
y
0has a maximum at (
t
0
,
x
0
), set (24)
φ
r
,
1
ε
t
,
x
=
v
r
t
,
y
0
+
1
ε
2
x
-
y
0
2
+
ε
t
-
l
2
,then w
r
(
t
,
x
)
-
φ
r
,
1
ε
(
t
,
x
), has a maximum at (
t
0
,
x
0
), and hence(25)
∂
w
r
-
φ
r
,
1
ε
∂
t
t
0
,
x
0
=
∂
w
r
-
φ
r
,
1
ε
∂
x
t
0
,
x
0
=
0
⟹
∂
φ
r
,
1
ε
∂
t
t
0
,
x
0
=
∂
w
r
∂
t
t
0
,
x
0
,
∂
φ
r
,
1
ε
∂
x
t
0
,
x
0
=
∂
w
r
∂
x
t
0
,
x
0
,
∂
Ψ
r
,
1
ε
∂
x
t
0
,
x
0
=
0
⟺
∂
w
r
∂
x
t
0
,
x
0
-
2
ε
2
x
0
-
y
0
=
0
,
∂
Ψ
r
,
1
ε
∂
t
t
0
,
x
0
=
0
⟺
∂
w
r
∂
t
t
0
,
x
0
-
∂
v
r
t
0
,
x
0
∂
t
t
0
,
x
0
-
2
ε
t
0
-
l
=
0
,
(26)
0
≤
∂
φ
r
,
1
ε
t
0
,
x
0
∂
t
-
H
r
t
0
,
x
0
,
2
ε
2
x
0
-
y
0
,
D
x
2
φ
r
,
1
ε
t
0
,
x
0
-
∑
k
≠
r
q
r
k
w
r
t
0
,
x
0
-
w
k
t
0
,
x
0
.
(ii) Sincev is a viscosity supersolution of (8) and the function(27)
Ψ
r
,
2
ε
:
x
,
t
⟼
-
ϕ
r
ε
t
,
x
0
,
yhas a minimum at (
t
0
,
y
0
), set (28)
φ
r
,
2
ε
t
,
y
=
w
r
t
,
x
0
-
1
ε
2
x
0
-
y
2
-
ε
t
-
l
2then φ
r
,
2
ε
(
t
,
x
)
-
v
r
(
t
,
x
), has a maximum at (
t
0
,
x
0
), and hence(29)
∂
φ
r
,
2
ε
-
v
r
∂
t
t
0
,
y
0
=
∂
φ
r
,
2
ε
-
v
r
∂
x
t
0
,
y
0
=
0
⟹
∂
φ
r
,
2
ε
∂
t
t
0
,
y
0
=
∂
v
r
∂
t
t
0
,
y
0
,
∂
φ
r
,
2
ε
∂
x
t
0
,
y
0
=
∂
v
r
∂
x
t
0
,
y
0
,
∂
Ψ
r
,
2
ε
∂
y
t
0
,
y
0
=
0
⟺
+
∂
v
r
∂
y
t
0
,
y
0
-
2
ε
2
x
0
-
y
0
=
0
,
∂
Ψ
r
,
2
ε
∂
t
t
0
,
x
0
=
0
⟺
-
∂
w
r
∂
t
t
0
,
x
0
+
∂
v
r
t
0
,
x
0
∂
t
t
0
,
x
0
+
2
ε
t
0
-
l
=
0
,
(30)
∂
φ
r
,
2
ε
t
0
,
y
0
∂
t
-
H
r
t
0
,
x
0
,
2
ε
2
x
0
-
y
0
,
D
y
2
φ
r
,
2
ε
t
0
,
y
0
-
∑
k
≠
r
q
r
k
v
r
t
0
,
y
0
-
v
k
t
0
,
y
0
≤
0
.By combining (26) and (30),(31)
0
≤
∂
w
r
t
0
,
x
0
∂
t
-
∂
v
r
t
0
,
y
0
∂
t
+
H
r
t
0
,
y
0
,
2
ε
2
x
0
-
y
0
,
D
y
2
φ
r
,
2
ε
t
0
,
y
0
-
H
r
t
0
,
x
0
,
2
ε
2
x
0
-
y
0
,
D
x
2
φ
r
,
1
ε
t
0
,
x
0
-
∑
k
≠
r
q
r
k
w
r
t
0
,
x
0
-
w
k
t
0
,
x
0
+
∑
k
≠
r
q
r
k
v
r
t
0
,
y
0
-
v
k
t
0
,
y
0
,and by Lemma 2
(32)
0
≤
2
ε
t
0
-
l
+
ω
2
ε
2
x
0
-
y
0
2
+
x
0
-
y
0
-
∑
k
≠
r
q
r
k
w
r
t
0
,
x
0
-
w
k
t
0
,
x
0
+
∑
k
≠
r
q
r
k
v
r
t
0
,
y
0
-
v
k
t
0
,
y
0
,we obtain (33)
0
≤
2
ε
t
0
-
l
+
ω
2
ε
2
x
0
-
y
0
2
+
x
0
-
y
0
-
∑
k
≠
r
q
r
k
w
r
t
0
,
x
0
-
v
r
t
0
,
y
0
+
∑
k
≠
r
q
r
k
w
k
t
0
,
x
0
-
v
k
t
0
,
y
0
.We have -
q
r
k
<
0 and w
r
(
t
0
,
x
0
)
-
v
r
(
t
0
,
y
0
)
>
0 (19) and hence (34)
0
≤
2
ε
t
0
-
l
+
ω
2
ε
2
x
0
-
y
0
2
+
x
0
-
y
0
+
∑
k
≠
r
q
r
k
w
k
t
0
,
x
0
-
v
k
t
0
,
y
0
.Since w
k
(
t
0
,
x
0
)
-
v
k
(
t
0
,
y
0
)
≤
w
r
(
t
0
,
x
0
)
-
v
r
(
t
0
,
y
0
) and ∑
k
≠
r
‍
q
r
k
=
-
q
k
k, we obtain (35)
0
≤
2
ε
t
0
-
l
+
ω
2
ε
2
x
0
-
y
0
2
+
x
0
-
y
0
-
q
k
k
w
r
t
0
,
x
0
-
v
r
t
0
,
y
0
.Thus, (36)
w
s
l
,
z
-
v
s
l
,
z
≤
ϕ
r
ε
t
0
,
x
0
,
y
0
≤
w
r
t
0
,
x
0
-
v
r
t
0
,
y
0
≤
1
q
k
k
2
ε
t
0
-
l
+
1
q
k
k
ω
2
ε
2
x
0
-
y
0
2
+
x
0
-
y
0
.To finish the proof, we need to show (37)
ω
2
ε
2
x
0
-
y
0
2
+
x
0
-
y
0
⟶
0
,
a
s
ε
⟶
0
.Let (38)
h
q
=
Sup
⁡
v
r
t
,
x
-
v
r
t
,
y
:
t
∈
0
,
T
,
x
,
y
∈
Q
¯
x
-
y
2
≤
q
,so that for any (
t
,
x
) and (
t
,
y
)
∈
Q
¯
(39)
v
r
t
,
x
-
v
r
t
,
y
≤
h
x
-
y
2
.Since (
x
0
,
t
0
,
y
0
,
r
) maximizes ϕ
r
ε over Q, (40)
w
r
t
0
,
x
0
-
v
r
t
0
,
y
0
-
1
ε
2
x
0
-
y
0
2
-
ε
t
0
-
l
2
≥
w
r
t
0
,
x
0
-
v
r
t
0
,
x
0
-
1
ε
2
x
0
-
x
0
2
-
ε
t
0
-
l
2
.We obtain (41)
1
ε
2
x
0
-
y
0
2
≤
v
r
t
0
,
x
0
-
v
r
t
0
,
y
0
≤
h
x
0
-
y
0
2
.Since h is bounded by some constant K, this implies that (42)
2
ε
2
x
0
-
y
0
2
≤
K
.The definition of h yields (43)
2
ε
2
x
0
-
y
0
2
+
x
0
-
y
0
≤
h
K
ε
2
2
+
ε
K
2
,and we obtain (44)
ω
2
ε
2
x
0
-
y
0
2
+
x
0
-
y
0
≤
ω
h
K
ε
2
2
+
ε
K
2
,and we obtain (45)
w
s
l
,
z
-
v
s
l
,
z
≤
0
a
s
ε
⟶
0
,which is a contradiction to (16).Corollary 4.
The viscosity solution satisfying the boundary condition is unique.Proof.
Ifv
k
1 and v
k
2 are 2 viscosity solutions such that v
k
1
=
v
k
2 on ∂
Q, then(i)
v
k
1
≤
v
k
2 on ∂
Q
⇒
v
k
1
≤
v
k
2 on Q accordingly (Proposition 3);
(ii)
v
k
1
≥
v
k
2 on ∂
Q
⇒
v
k
1
≥
v
k
2 on Q accordingly (Proposition 3).Hence v
k
1
=
v
k
2 on Q.


## 4. Application in Finance: Continuous-Time Mean-Variance Model without Shorting where the Market Parameters Are Random

We now briefly recall the results of the continuous-time mean-variance model without shorting [5] and the mean-variance portfolio selection problem in continuous time where the market parameters are random processes [4].We study the intersection of the both cases [4, 5], that is, continuous-time mean-variance model without shorting where the market parameters are random.Consider a market in whichn
+
1 assets are traded continuously on a finite time horizon [
0
,
T
]. One of the assets is a bank account whose price P
0
(
t
) is subject to the stochastic ODE (ordinary differential equation)(46)
d
P
0
t
=
r
t
,
α
t
P
0
t
d
t
,
t
∈
0
,
T
P
0
0
=
p
0
>
0
,
t
∈
0
,
T
,
α
t
=
i
∈
N
,where r
(
t
,
i
)
≥
0, i
=
1,2
,
…
,
m, are given as interest rate processes corresponding to different market modes. The other n assets are stocks whose price processes P
m
(
t
)
m
=
1,2
,
…
,
n satisfy the system of SDE (system of differential equation)(47)
d
P
m
t
=
P
m
t
b
m
t
,
α
t
d
t
+
∑
p
=
1
n
σ
m
p
t
,
α
t
d
W
p
t
,
t
∈
0
,
T
P
m
0
=
p
m
>
0
,
t
∈
0
,
T
,
α
t
=
i
∈
N
,where for each i
=
1,2
,
…
,
n
b
m
(
t
,
i
) is the appreciation rate process and σ
m
(
t
,
i
)
=
(
σ
m
1
(
t
,
i
)
,
…
,
σ
m
n
(
t
,
i
)
) is the volatility or the dispersion rate process of the mth stock, corresponding to α
(
t
)
=
i.Define the volatility matrix(48)
σ
t
,
i
≡
σ
m
p
t
,
i
n
×
n
for  each
i
=
1
,
…
,
n
.We assume(49)
σ
t
,
i
σ
t
,
i
′
≥
δ
I
∀
t
∈
0
,
T
,
δ
>
0and r
(
t
,
i
), b
m
(
t
,
i
), σ
m
n
(
t
,
i
) are measurable and uniformly bounded in t.Denote byy
(
t
) the total wealth of the agent with y
(
0
)
=
y
0
>
0 being his initial wealth; y
(
t
) satisfies(50)
d
y
t
=
r
t
,
i
y
t
+
B
t
,
i
u
t
+
∑
m
=
1
n
b
m
t
,
i
-
r
t
,
i
u
m
t
d
t
+
∑
p
=
1
n
∑
m
=
1
n
σ
m
p
t
,
i
u
m
t
d
W
p
t
,
t
∈
s
,
T
y
0
=
y
0
>
0
,
α
0
=
i
0
,
t
h
e
i
n
i
t
i
a
l
m
a
r
k
e
t
m
o
d
e
,where u
m
(
t
) is the total market value of the agent’s wealth in the mth asset and m
=
0,1
,
…
,
n at time t.u
(
·
)
=
u
1
·
,
…
,
u
n
·
′ is called a portfolio of the agent.u
0
(
·
), the asset in the bank account, is completely specified since u
0
(
t
)
=
y
(
t
)
-
∑
i
=
1
n
‍
u
i
(
t
)
. Thus, in our analysis to follow, only u
(
·
) is considered.Setting(51)
B
t
,
i
=
b
1
t
,
i
-
r
t
,
i
,
…
,
b
n
t
,
i
-
r
t
,
i
,
i
∈
M
,wealth equation (50) satisfies(52)
d
y
t
=
r
t
,
i
y
t
+
B
t
,
i
u
t
d
t
+
u
t
′
σ
t
,
i
d
W
t
,
t
∈
s
,
T
y
0
=
y
0
>
0
,
α
0
=
i
0
.The objective of the agent is to find an admissible portfolio u
(
·
)
≥
0, whose expected terminal wealth is E
t
y
y
(
T
)
=
d for a given d
∈
R, so that the risk is measured by the variance of the terminal wealth. Namely, the goal of the agent is to solve the following constrained stochastic optimization problem, parameterized by d
∈
R:(53)
m
i
n
i
m
i
z
e
J
M
V
y
0
,
i
0
,
u
·
=
E
t
y
y
T
-
d
2
,
s
u
b
j
e
c
t
t
o
E
t
y
y
T
=
d
,
y
·
,
u
·
a
d
m
i
s
s
i
b
l
e
,called mean-variance portfolio.Formula (53) is a convex optimization problem; by using a Lagrange multiplier μ
∈
R, we can attach the equality constraint E
t
y
y
(
T
)
=
d to the first equation of (53). In this way, the portfolio problem can be solved via the following optimal stochastic control problem:P
(
d
):(54)
m
i
n
i
m
i
z
e
E
t
y
y
T
-
d
2
+
2
μ
E
t
y
y
T
-
d
,
s
u
b
j
e
c
t
t
o
E
t
y
y
T
=
d
,
y
·
,
u
·
a
d
m
i
s
s
i
b
l
e
,where factor 2 in front of the multiplier μ is introduced in the objective function just for convenience.This problem is equivalent to the following:(
A
(
μ
)
):(55)
m
i
n
i
m
i
z
e
E
t
y
1
2
y
T
-
d
-
μ
2
,
s
u
b
j
e
c
t
t
o
u
·
∈
L
F
2
0
,
T
;
R
+
m
y
·
,
u
·
a
d
m
i
s
s
i
b
l
e
,in the sense that two problems have exactly the same optimal control [5].Next, we letx
(
t
)
=
y
(
t
)
-
(
d
-
μ
).Consider(
A
(
μ
)
):(56)
m
i
n
i
m
i
z
e
E
t
x
1
2
x
T
2
,
s
u
b
j
e
c
t
t
o
u
·
∈
L
F
2
0
,
T
;
R
+
m
x
·
,
u
·
a
d
m
i
s
s
i
b
l
e
,and (52) is equivalent to(57)
d
x
t
=
A
t
,
i
x
t
+
B
t
,
i
u
t
+
A
t
,
i
d
-
μ
d
t
+
∑
p
=
1
n
G
p
t
,
i
u
t
d
W
p
t
,
t
∈
s
,
T
x
s
=
y
s
-
d
-
μ
∈
R
w
h
e
r
e
G
p
t
,
i
=
σ
1
p
t
,
i
,
…
,
σ
n
p
t
,
i
A
t
,
i
∈
R
.Problem A
(
μ
) is a stochastic optimal linear quadratic coupled (LQC) control problem, and we can get the solution of (
A
(
μ
)
) by guessing the solution as a quadratic function. By making use of the duality relationship between (
P
(
d
)
) and (
A
(
μ
)
), see Appendix A.2; we obtain the solution of the original problem (
P
(
d
)
).


### 4.1. A General Constrained Stochastic Linear Quadratic Problem

Consider controlled linear stochastic differential equation (57).We assume that the matrix∑
p
=
1
n
‍
G
p
t
,
i
′
G
p
(
t
,
i
) is nonsingular. Our objective is to find an optimal control u
(
·
) that minimizes the quadratic terminal cost function. Set (58)
U
s
,
T
=
L
F
2
s
,
T
;
R
+
m
.Given u
(
·
)
∈
U
[
s
,
T
], the pair (
x
(
·
)
,
u
(
·
)
) is admissible if x
(
·
)
∈
L
F
2
(
s
,
T
;
R
) is a solution of (57). Let(59)
J
i
s
,
x
;
u
·
=
E
t
x
1
2
x
T
2
.The value function associated with LQC problem (57) and (59) is defined by(60)
V
i
s
,
x
=
inf
u
·
∈
U
s
,
T
⁡
J
i
s
,
x
;
u
·
.In Appendix A.3, and also [8], value function (60) satisfies (8). Next, we will provide an explicit viscosity solution of (8).Definition 5.
(i)
A portfoliou
(
·
) is said to be admissible if u
(
·
)
∈
L
F
2
(
0
,
T
;
R
+
n
) and the SDE (57) has a unique solution x
(
·
) corresponding to u
(
·
). In this case, we refer to (
x
(
·
)
,
u
(
·
)
) as an admissible (wealth, portfolio) pair.
(ii)
The problem is calledfeasible if there is at least one portfolio satisfying all the constraints.
(iii)
The problem is called finite if it isfeasible and the infimum of J
M
V
(
x
0
,
i
0
,
u
(
·
)
) is finite.
(iv)
An optimal portfolio to the above problem, if it ever exists, is called anefficient portfolio corresponding to
d, and the corresponding (
V
a
r
x
(
T
)
,
d
)
∈
R
2and (
σ
x
(
T
)
,
d
)
∈
R
2
) are interchangeably called an efficient point, and the set of all the efficient points is called the efficient frontier.Next, we let(61)
f
¯
i
t
,
x
;
u
=
A
t
,
i
x
t
+
B
t
,
i
u
+
f
t
,
i
,
g
¯
i
t
,
x
;
u
=
G
t
,
i
u
,
w
h
e
r
e
f
t
,
i
=
A
t
,
i
d
-
μ
.


### 4.2. Viscosity Solution of the Coupled System

By guessing the value function of (8) as(62)
V
i
t
,
x
=
1
2
P
t
,
i
x
2
+
M
t
,
i
x
+
R
t
,
iwe will see that the coefficients of (8) satisfy the following Riccati equation.Definition 6.
We define the system of Riccati equations as follows(63)
d
P
^
t
,
i
d
t
=
-
2
A
t
,
i
-
ξ
¯
t
,
i
2
-
2
B
i
t
G
-
1
t
,
i
ξ
¯
t
,
i
P
^
t
,
i
+
∑
j
≠
i
n
q
i
,
j
P
^
t
,
i
-
P
^
t
,
j
,
P
^
T
,
k
=
1
,
(64)
d
M
^
t
,
i
d
t
=
-
A
t
,
i
-
ξ
¯
t
,
i
2
-
2
B
t
,
i
G
-
1
t
,
i
ξ
¯
t
,
i
M
^
t
,
i
-
P
^
t
,
i
f
t
,
i
+
∑
j
≠
i
n
q
i
,
j
M
^
t
,
i
-
M
^
t
,
j
,
M
^
T
,
k
=
0
,
(65)
d
R
^
t
,
i
d
t
=
-
M
^
t
,
i
f
t
,
i
-
B
t
,
i
G
-
1
t
,
i
ξ
¯
t
,
i
M
^
2
t
,
i
P
^
t
,
i
-
1
-
1
2
ξ
¯
t
,
i
2
M
^
2
t
,
i
P
^
t
,
i
-
1
+
∑
j
≠
i
n
q
i
,
j
R
^
t
,
i
-
R
^
t
,
j
,
R
^
T
,
k
=
0
,
(66)
d
P
~
t
,
i
d
t
=
-
2
A
t
,
i
P
~
t
,
i
+
∑
j
≠
i
n
q
i
,
j
P
^
t
,
i
-
P
^
t
,
j
,
P
^
T
,
k
=
1
,
(67)
d
M
~
t
,
i
d
t
=
-
A
t
,
i
M
~
t
,
i
+
P
~
t
,
i
f
t
,
i
+
∑
j
≠
i
n
q
i
,
j
M
~
t
,
i
-
M
~
t
,
j
,
M
^
T
,
k
=
0
,
(68)
d
R
~
t
,
i
d
t
=
-
M
~
t
,
i
f
t
,
i
+
∑
j
≠
i
n
q
i
,
j
R
~
t
,
i
-
R
~
t
,
j
,
R
^
T
,
k
=
0
,where ξ
¯
(
t
,
i
) is as in Lemma A.1Remark 7.
By letting(69)
α
~
t
,
i
=
-
2
A
t
,
i
-
ξ
¯
t
,
i
2
-
2
B
t
,
i
G
-
1
t
,
i
ξ
¯
t
,
i
,
β
~
t
,
i
=
-
2
A
t
,
i
-
ξ
¯
t
,
i
2
-
2
B
t
,
i
G
-
1
t
,
i
ξ
¯
t
,
i
,
γ
~
t
,
i
=
-
M
^
t
,
i
f
t
,
i
-
B
t
,
i
G
-
1
t
,
i
ξ
¯
t
,
i
M
^
2
t
,
i
P
^
t
,
i
-
1
2
ξ
¯
t
,
i
2
M
^
2
t
,
i
P
^
t
,
i
.We see that (63) is equivalent to(70)
d
P
^
t
d
t
=
M
t
P
^
t
,
M
t
=
α
~
t
,
i
δ
i
j
+
q
i
j
1
≤
i
,
j
≤
n
,
P
^
T
=
1
,
w
h
e
r
e
P
^
t
=
P
^
t
,
i
1
≤
i
≤
m
;(64) is equivalent to(71)
d
M
^
t
d
t
=
N
t
M
^
t
+
G
t
,
N
t
=
β
~
t
,
i
δ
i
j
+
q
i
j
1
≤
i
,
j
≤
n
,
G
t
=
P
^
1
f
1
t
⋯
P
^
n
f
n
t
,
M
^
T
=
0
,
w
h
e
r
e
M
^
t
=
M
^
t
,
i
1
≤
i
≤
m
;(65) is equivalent to(72)
d
R
^
t
d
t
=
Q
t
R
^
t
+
γ
~
t
,
Q
t
=
q
i
j
1
≤
i
,
j
≤
n
,
R
^
T
=
0
,
w
h
e
r
e
R
^
t
=
R
^
t
,
i
1
≤
i
≤
m
;(66) is equivalent to(73)
d
P
~
t
d
t
=
H
t
P
~
t
,
H
t
=
-
2
A
i
t
δ
i
j
+
q
i
j
1
≤
i
,
j
≤
n
,
P
~
T
=
0
,
w
h
e
r
e
P
~
t
=
P
~
t
,
i
1
≤
i
≤
m
;(67) is equivalent to(74)
d
M
~
t
d
t
=
L
t
M
~
t
+
K
t
,
L
t
=
-
A
i
t
δ
i
j
+
q
i
j
1
≤
i
,
j
≤
n
,
K
t
=
P
~
1
f
1
t
⋯
P
~
n
f
n
t
,
P
~
T
=
0
,
w
h
e
r
e
M
~
t
=
M
~
t
,
i
1
≤
i
≤
m
;(68) is equivalent to(75)
d
R
~
t
d
t
=
Q
t
R
~
t
+
O
t
,
Q
t
=
q
i
j
1
≤
i
,
j
≤
n
,
O
t
=
M
~
1
f
1
t
⋯
M
~
n
f
n
t
,
R
~
t
=
0
,
w
h
e
r
e
R
~
t
=
R
~
t
,
i
1
≤
i
≤
m
.


### 4.3. Riccati Equation Magnus Approach

We will show how to provide the solutions of (70)–(75) by making use of Magnus method.Proposition 8 (see [10]).
Given then
×
n coefficient matrix A
(
t
), (76)
d
Y
t
d
t
=
A
t
Y
t
,
Y
t
0
=
Y
0
w
h
e
r
e
Y
t
=
Y
t
,
i
1
≤
i
≤
mand then Y
(
t
)
=
e
x
p
Ω
t
,
t
0
Y
0 which is subsequently constructed as a series expansion (77)
Ω
t
,
t
0
=
∑
k
=
1
∞
Ω
k
t
,
t
0
w
h
e
r
e
Ω
1
=
∫
t
0
t
A
τ
d
τ
Ω
n
t
,
t
0
=
∑
j
=
1
n
-
1
B
j
j
!
∫
t
0
t
S
n
j
τ
d
τ
,
n
≥
2
,where S
n
j is defined recursively by (78)
S
n
j
=
∑
m
=
1
n
-
j
Ω
m
,
S
n
-
m
j
-
1
,
2
≤
j
≤
n
-
1
,
S
n
1
=
Ω
n
-
1
,
A
,
S
n
n
-
1
=
a
d
Ω
1
n
-
1
A
,
a
d
Ω
k iterated commutator(79)
a
d
Ω
A
=
Ω
,
A
,
a
d
Ω
k
+
1
A
=
Ω
,
a
d
Ω
k
A
,
a
d
Ω
0
A
=
A
,
k
∈
N
,and B
j is the Bernoulli numbers.Proposition 99 (see [11]).
(80)
d
Y
t
d
t
=
M
t
Y
t
+
Y
t
N
t
+
F
t
,
Y
t
0
=
Y
0
t
∈
t
0
,
T
,where Y
(
t
)
,
F
(
t
)
∈
C
p
×
q
,
M
(
t
)
∈
C
p
×
p, and N
(
t
)
∈
C
q
×
q. The solution of (80) is given by (81)
Y
t
=
Φ
M
t
,
t
0
Y
0
Φ
N
∗
t
,
t
0
+
Ψ
t
,
t
0with(82)
Ψ
t
,
t
0
=
∫
t
0
t
Ψ
M
t
,
s
F
s
Φ
N
∗
t
,
s
d
s
,where Φ
m
(
t
,
t
0
) and Φ
N
∗
(
t
,
t
0
) are the fundamental solution matrices of the associated homogeneous equations (83)
Φ
M
′
t
,
t
0
=
M
t
Φ
M
t
,
t
0
,
Φ
M
t
0
,
t
0
=
I
p
,
Φ
∗
′
t
,
t
0
=
Φ
∗
t
,
t
0
N
t
,
Φ
N
t
0
,
t
0
=
I
q
.Remark 10.
By making use of Proposition8 we get (70) and (71)–(75) are special case of Proposition 9 when N
=
[
0
i
j
]
n
×
n.Theorem 11.
The value function of (60) is given by(84)
V
i
t
,
x
=
V
¯
i
t
,
x
=
1
2
P
^
t
,
i
x
2
+
M
^
t
,
i
x
+
R
^
t
,
i
,
i
f
x
+
η
¯
t
,
i
≤
0
,
V
~
i
t
,
x
=
1
2
P
~
t
,
i
x
2
+
M
~
t
,
i
x
+
R
~
t
,
i
,
i
f
x
+
η
¯
t
,
i
>
0and the optimal control is given by(85)
u
∗
=
-
G
t
,
i
′
-
1
ξ
¯
t
,
i
x
+
η
¯
t
,
i
,
i
f
x
+
η
¯
t
,
i
≤
0
,
0
,
i
f
x
+
η
¯
t
,
i
>
0
,where (86)
η
¯
t
,
i
=
M
^
t
,
i
R
^
t
,
i
.Proof.
Let(87)
Γ
1
i
=
t
,
x
,
i
∈
0
,
T
×
R
×
N
∣
x
+
η
¯
t
,
i
≤
0
,
Γ
2
i
=
t
,
x
,
i
∈
0
,
T
×
R
∣
x
+
η
¯
t
,
i
>
0
.
(i)
InΓ
1
i, V as given by (62) is well defined, with (88)
∂
V
i
t
,
x
∂
t
=
1
2
P
^
˙
t
,
i
x
2
+
M
^
˙
t
,
i
x
+
R
^
˙
t
,
i
,
∂
V
i
t
,
x
∂
x
=
P
^
t
,
i
x
+
M
^
t
,
i
,
∂
2
V
i
t
,
x
∂
x
2
=
P
^
t
,
i
.
Substituting them into the left-hand side (LHS) of (8), we obtain(89)
L
H
S
=
1
2
P
^
˙
t
,
i
+
P
^
t
,
i
A
t
+
1
2
∑
j
≠
i
q
i
j
P
^
t
,
i
-
P
^
t
,
j
x
2
M
^
˙
t
,
i
+
P
^
t
,
i
f
t
,
i
+
M
^
t
,
i
A
i
t
+
∑
j
≠
i
q
i
j
R
^
t
,
i
-
R
^
t
,
i
x
+
R
^
˙
t
,
i
+
M
^
t
,
i
f
t
,
i
+
∑
j
≠
i
q
i
j
R
^
t
,
i
-
R
^
t
,
i
inf
u
≥
0
⁡
1
2
u
′
G
t
,
i
′
G
t
,
i
u
+
B
t
,
i
x
+
M
^
t
,
i
P
^
t
,
i
u
.
Letη
¯
(
t
,
i
)
=
M
^
(
t
,
i
)
/
P
^
(
t
,
i
) and, by using Lemma A.1  α
¯
=
-
[
x
+
η
¯
(
t
,
i
)
]
>
0, it follows that the minimizer of (89) is achieved by(90)
u
∗
=
-
G
t
,
i
′
-
1
ξ
¯
t
,
i
x
+
η
¯
t
,
i
.
Substitutingu
∗
(
t
,
x
) back into (8) and noting (63)–(65), it immediately follows that L
H
S
=
0
.
Now, we will show thatV
¯ is a viscosity subsolution.
Letφ
i
∈
C
2
(
Q
) and choose (
t
¯
,
x
¯
)
∈
a
r
g
m
a
x
{
(
V
¯
i
-
φ
i
)
(
t
,
x
)
∣
(
t
,
x
)
∈
Q
¯
}
∩
Q; then, (91)
∂
φ
i
t
¯
,
x
¯
∂
t
=
∂
V
¯
i
t
¯
,
x
¯
∂
t
,
∂
φ
i
t
¯
,
x
¯
∂
x
=
∂
V
¯
i
t
¯
,
x
¯
∂
x
,
∂
2
V
¯
i
-
φ
i
∂
x
2
t
¯
,
x
¯
≤
0
⟹
P
¯
i
=
∂
2
V
¯
i
t
¯
,
x
¯
∂
x
2
≤
∂
2
φ
i
t
¯
,
x
¯
∂
x
2
and we obtain(92)
0
=
∂
t
V
¯
i
t
¯
,
x
¯
+
inf
u
≥
0
⁡
1
2
g
¯
i
t
¯
,
x
¯
,
u
g
¯
i
t
¯
,
x
¯
,
u
′
D
x
2
V
¯
i
t
¯
,
x
¯
+
f
¯
i
t
¯
,
x
¯
,
u
D
x
V
¯
i
t
¯
,
x
¯
-
∑
j
≠
i
q
i
j
V
¯
i
t
¯
,
x
¯
-
V
¯
j
t
¯
,
x
¯
≤
∂
t
φ
i
t
¯
,
x
¯
+
inf
u
≥
0
⁡
1
2
g
¯
i
t
¯
,
x
¯
,
u
g
¯
i
t
¯
,
x
¯
,
u
′
D
x
2
φ
i
t
¯
,
x
¯
+
f
¯
i
t
¯
,
x
¯
,
u
φ
i
t
¯
,
x
¯
-
∑
j
≠
i
q
i
j
V
¯
i
t
¯
,
x
¯
-
V
¯
j
t
¯
,
x
¯
.
Hence,V
¯
i is a viscosity subsolution.
(ii)
InΓ
2
i, we proceed similarly with (93)
∂
V
i
t
,
x
∂
t
=
1
2
P
~
˙
t
,
i
x
2
+
M
~
˙
t
,
i
x
+
R
~
˙
t
,
i
,
∂
V
i
t
,
x
∂
x
=
P
~
t
,
i
x
+
M
~
t
,
i
,
∂
2
V
i
t
,
x
∂
x
2
=
P
~
t
,
i
.
Substituting them into the left-hand side (LHS) of (8), we obtain(94)
L
H
S
=
1
2
P
~
˙
t
,
i
+
P
~
˙
t
,
i
A
t
,
i
+
1
2
∑
j
≠
i
q
i
j
P
~
˙
t
,
i
-
P
~
˙
t
,
j
x
2
M
~
˙
t
,
i
+
P
~
t
,
i
f
i
t
+
M
~
t
,
i
A
t
,
i
+
∑
j
≠
i
q
i
j
R
~
t
,
i
-
R
~
t
,
j
x
+
R
~
˙
t
,
i
+
M
~
˙
t
,
i
f
t
,
i
+
∑
j
≠
i
q
i
j
R
~
t
,
i
-
R
~
t
,
j
inf
u
≥
0
⁡
1
2
u
′
G
t
,
i
′
G
i
t
u
+
B
t
,
i
x
+
M
~
t
,
i
P
~
t
,
i
u
.
Sinceα
=
-
[
x
+
η
¯
(
t
,
i
)
]
>
0, the minimizer of (94) is (95)
u
i
∗
=
0
.
Substitutingu
∗ into (8), it is easy to show that V
~ satisfies HJBC equation (8) in Γ
2
i.
Now, we will show thatV
~ is a viscosity subsolution.
Letϕ
i
∈
C
2
(
Q
) and choose (
t
~
,
x
~
)
∈
argmin
⁡
{
(
V
~
i
-
φ
i
)
(
t
,
x
)
∣
(
t
,
x
)
∈
Q
¯
}
∩
Q; then, (96)
∂
ϕ
i
t
~
,
x
~
∂
t
=
∂
V
~
i
t
~
,
x
~
∂
t
,
∂
ϕ
i
t
~
,
x
~
∂
x
=
∂
V
~
i
t
~
,
x
~
∂
x
,
∂
2
V
~
i
-
ϕ
i
∂
x
2
t
~
,
x
~
≥
0
⟹
P
~
i
=
∂
2
V
~
i
t
~
,
x
~
∂
x
2
≥
∂
2
ϕ
i
t
~
,
x
~
∂
x
2
and we obtain(97)
0
=
∂
t
V
~
i
t
~
,
x
~
+
inf
u
≥
0
⁡
1
2
g
¯
i
t
~
,
x
~
,
u
g
¯
i
t
~
,
x
~
,
u
′
D
x
2
V
~
i
t
~
,
x
~
+
f
¯
i
t
~
,
x
~
,
u
D
x
V
~
i
t
~
,
x
~
-
∑
j
≠
i
q
i
j
V
~
i
t
~
,
x
~
-
V
~
j
t
~
,
x
~
≥
∂
t
ϕ
i
t
~
,
x
~
+
inf
u
≥
0
⁡
1
2
g
¯
i
t
~
,
x
~
,
u
g
¯
i
t
~
,
x
~
,
u
′
D
x
2
ϕ
i
t
~
,
x
~
+
f
¯
i
t
~
,
x
~
,
u
ϕ
i
t
~
,
x
~
-
∑
j
≠
i
q
i
j
V
~
i
t
~
,
x
~
-
V
~
j
t
~
,
x
~
.
Hence,V
~
i is a viscosity supersolution.We see that the value function V is a viscosity solution.
Remark  12. we see clearly that ∂
2
V
(
t
,
x
,
i
)
/
∂
x
2 does not exist in Q, since P
~
(
t
,
i
)
≠
P
^
(
t
,
i
). For this reason, we are required to work within the framework of viscosity solutions.


## 4.1. A General Constrained Stochastic Linear Quadratic Problem

Consider controlled linear stochastic differential equation (57).We assume that the matrix∑
p
=
1
n
‍
G
p
t
,
i
′
G
p
(
t
,
i
) is nonsingular. Our objective is to find an optimal control u
(
·
) that minimizes the quadratic terminal cost function. Set (58)
U
s
,
T
=
L
F
2
s
,
T
;
R
+
m
.Given u
(
·
)
∈
U
[
s
,
T
], the pair (
x
(
·
)
,
u
(
·
)
) is admissible if x
(
·
)
∈
L
F
2
(
s
,
T
;
R
) is a solution of (57). Let(59)
J
i
s
,
x
;
u
·
=
E
t
x
1
2
x
T
2
.The value function associated with LQC problem (57) and (59) is defined by(60)
V
i
s
,
x
=
inf
u
·
∈
U
s
,
T
⁡
J
i
s
,
x
;
u
·
.In Appendix A.3, and also [8], value function (60) satisfies (8). Next, we will provide an explicit viscosity solution of (8).Definition 5.
(i)
A portfoliou
(
·
) is said to be admissible if u
(
·
)
∈
L
F
2
(
0
,
T
;
R
+
n
) and the SDE (57) has a unique solution x
(
·
) corresponding to u
(
·
). In this case, we refer to (
x
(
·
)
,
u
(
·
)
) as an admissible (wealth, portfolio) pair.
(ii)
The problem is calledfeasible if there is at least one portfolio satisfying all the constraints.
(iii)
The problem is called finite if it isfeasible and the infimum of J
M
V
(
x
0
,
i
0
,
u
(
·
)
) is finite.
(iv)
An optimal portfolio to the above problem, if it ever exists, is called anefficient portfolio corresponding to
d, and the corresponding (
V
a
r
x
(
T
)
,
d
)
∈
R
2and (
σ
x
(
T
)
,
d
)
∈
R
2
) are interchangeably called an efficient point, and the set of all the efficient points is called the efficient frontier.Next, we let(61)
f
¯
i
t
,
x
;
u
=
A
t
,
i
x
t
+
B
t
,
i
u
+
f
t
,
i
,
g
¯
i
t
,
x
;
u
=
G
t
,
i
u
,
w
h
e
r
e
f
t
,
i
=
A
t
,
i
d
-
μ
.


## 4.2. Viscosity Solution of the Coupled System

By guessing the value function of (8) as(62)
V
i
t
,
x
=
1
2
P
t
,
i
x
2
+
M
t
,
i
x
+
R
t
,
iwe will see that the coefficients of (8) satisfy the following Riccati equation.Definition 6.
We define the system of Riccati equations as follows(63)
d
P
^
t
,
i
d
t
=
-
2
A
t
,
i
-
ξ
¯
t
,
i
2
-
2
B
i
t
G
-
1
t
,
i
ξ
¯
t
,
i
P
^
t
,
i
+
∑
j
≠
i
n
q
i
,
j
P
^
t
,
i
-
P
^
t
,
j
,
P
^
T
,
k
=
1
,
(64)
d
M
^
t
,
i
d
t
=
-
A
t
,
i
-
ξ
¯
t
,
i
2
-
2
B
t
,
i
G
-
1
t
,
i
ξ
¯
t
,
i
M
^
t
,
i
-
P
^
t
,
i
f
t
,
i
+
∑
j
≠
i
n
q
i
,
j
M
^
t
,
i
-
M
^
t
,
j
,
M
^
T
,
k
=
0
,
(65)
d
R
^
t
,
i
d
t
=
-
M
^
t
,
i
f
t
,
i
-
B
t
,
i
G
-
1
t
,
i
ξ
¯
t
,
i
M
^
2
t
,
i
P
^
t
,
i
-
1
-
1
2
ξ
¯
t
,
i
2
M
^
2
t
,
i
P
^
t
,
i
-
1
+
∑
j
≠
i
n
q
i
,
j
R
^
t
,
i
-
R
^
t
,
j
,
R
^
T
,
k
=
0
,
(66)
d
P
~
t
,
i
d
t
=
-
2
A
t
,
i
P
~
t
,
i
+
∑
j
≠
i
n
q
i
,
j
P
^
t
,
i
-
P
^
t
,
j
,
P
^
T
,
k
=
1
,
(67)
d
M
~
t
,
i
d
t
=
-
A
t
,
i
M
~
t
,
i
+
P
~
t
,
i
f
t
,
i
+
∑
j
≠
i
n
q
i
,
j
M
~
t
,
i
-
M
~
t
,
j
,
M
^
T
,
k
=
0
,
(68)
d
R
~
t
,
i
d
t
=
-
M
~
t
,
i
f
t
,
i
+
∑
j
≠
i
n
q
i
,
j
R
~
t
,
i
-
R
~
t
,
j
,
R
^
T
,
k
=
0
,where ξ
¯
(
t
,
i
) is as in Lemma A.1Remark 7.
By letting(69)
α
~
t
,
i
=
-
2
A
t
,
i
-
ξ
¯
t
,
i
2
-
2
B
t
,
i
G
-
1
t
,
i
ξ
¯
t
,
i
,
β
~
t
,
i
=
-
2
A
t
,
i
-
ξ
¯
t
,
i
2
-
2
B
t
,
i
G
-
1
t
,
i
ξ
¯
t
,
i
,
γ
~
t
,
i
=
-
M
^
t
,
i
f
t
,
i
-
B
t
,
i
G
-
1
t
,
i
ξ
¯
t
,
i
M
^
2
t
,
i
P
^
t
,
i
-
1
2
ξ
¯
t
,
i
2
M
^
2
t
,
i
P
^
t
,
i
.We see that (63) is equivalent to(70)
d
P
^
t
d
t
=
M
t
P
^
t
,
M
t
=
α
~
t
,
i
δ
i
j
+
q
i
j
1
≤
i
,
j
≤
n
,
P
^
T
=
1
,
w
h
e
r
e
P
^
t
=
P
^
t
,
i
1
≤
i
≤
m
;(64) is equivalent to(71)
d
M
^
t
d
t
=
N
t
M
^
t
+
G
t
,
N
t
=
β
~
t
,
i
δ
i
j
+
q
i
j
1
≤
i
,
j
≤
n
,
G
t
=
P
^
1
f
1
t
⋯
P
^
n
f
n
t
,
M
^
T
=
0
,
w
h
e
r
e
M
^
t
=
M
^
t
,
i
1
≤
i
≤
m
;(65) is equivalent to(72)
d
R
^
t
d
t
=
Q
t
R
^
t
+
γ
~
t
,
Q
t
=
q
i
j
1
≤
i
,
j
≤
n
,
R
^
T
=
0
,
w
h
e
r
e
R
^
t
=
R
^
t
,
i
1
≤
i
≤
m
;(66) is equivalent to(73)
d
P
~
t
d
t
=
H
t
P
~
t
,
H
t
=
-
2
A
i
t
δ
i
j
+
q
i
j
1
≤
i
,
j
≤
n
,
P
~
T
=
0
,
w
h
e
r
e
P
~
t
=
P
~
t
,
i
1
≤
i
≤
m
;(67) is equivalent to(74)
d
M
~
t
d
t
=
L
t
M
~
t
+
K
t
,
L
t
=
-
A
i
t
δ
i
j
+
q
i
j
1
≤
i
,
j
≤
n
,
K
t
=
P
~
1
f
1
t
⋯
P
~
n
f
n
t
,
P
~
T
=
0
,
w
h
e
r
e
M
~
t
=
M
~
t
,
i
1
≤
i
≤
m
;(68) is equivalent to(75)
d
R
~
t
d
t
=
Q
t
R
~
t
+
O
t
,
Q
t
=
q
i
j
1
≤
i
,
j
≤
n
,
O
t
=
M
~
1
f
1
t
⋯
M
~
n
f
n
t
,
R
~
t
=
0
,
w
h
e
r
e
R
~
t
=
R
~
t
,
i
1
≤
i
≤
m
.


## 4.3. Riccati Equation Magnus Approach

We will show how to provide the solutions of (70)–(75) by making use of Magnus method.Proposition 8 (see [10]).
Given then
×
n coefficient matrix A
(
t
), (76)
d
Y
t
d
t
=
A
t
Y
t
,
Y
t
0
=
Y
0
w
h
e
r
e
Y
t
=
Y
t
,
i
1
≤
i
≤
mand then Y
(
t
)
=
e
x
p
Ω
t
,
t
0
Y
0 which is subsequently constructed as a series expansion (77)
Ω
t
,
t
0
=
∑
k
=
1
∞
Ω
k
t
,
t
0
w
h
e
r
e
Ω
1
=
∫
t
0
t
A
τ
d
τ
Ω
n
t
,
t
0
=
∑
j
=
1
n
-
1
B
j
j
!
∫
t
0
t
S
n
j
τ
d
τ
,
n
≥
2
,where S
n
j is defined recursively by (78)
S
n
j
=
∑
m
=
1
n
-
j
Ω
m
,
S
n
-
m
j
-
1
,
2
≤
j
≤
n
-
1
,
S
n
1
=
Ω
n
-
1
,
A
,
S
n
n
-
1
=
a
d
Ω
1
n
-
1
A
,
a
d
Ω
k iterated commutator(79)
a
d
Ω
A
=
Ω
,
A
,
a
d
Ω
k
+
1
A
=
Ω
,
a
d
Ω
k
A
,
a
d
Ω
0
A
=
A
,
k
∈
N
,and B
j is the Bernoulli numbers.Proposition 99 (see [11]).
(80)
d
Y
t
d
t
=
M
t
Y
t
+
Y
t
N
t
+
F
t
,
Y
t
0
=
Y
0
t
∈
t
0
,
T
,where Y
(
t
)
,
F
(
t
)
∈
C
p
×
q
,
M
(
t
)
∈
C
p
×
p, and N
(
t
)
∈
C
q
×
q. The solution of (80) is given by (81)
Y
t
=
Φ
M
t
,
t
0
Y
0
Φ
N
∗
t
,
t
0
+
Ψ
t
,
t
0with(82)
Ψ
t
,
t
0
=
∫
t
0
t
Ψ
M
t
,
s
F
s
Φ
N
∗
t
,
s
d
s
,where Φ
m
(
t
,
t
0
) and Φ
N
∗
(
t
,
t
0
) are the fundamental solution matrices of the associated homogeneous equations (83)
Φ
M
′
t
,
t
0
=
M
t
Φ
M
t
,
t
0
,
Φ
M
t
0
,
t
0
=
I
p
,
Φ
∗
′
t
,
t
0
=
Φ
∗
t
,
t
0
N
t
,
Φ
N
t
0
,
t
0
=
I
q
.Remark 10.
By making use of Proposition8 we get (70) and (71)–(75) are special case of Proposition 9 when N
=
[
0
i
j
]
n
×
n.Theorem 11.
The value function of (60) is given by(84)
V
i
t
,
x
=
V
¯
i
t
,
x
=
1
2
P
^
t
,
i
x
2
+
M
^
t
,
i
x
+
R
^
t
,
i
,
i
f
x
+
η
¯
t
,
i
≤
0
,
V
~
i
t
,
x
=
1
2
P
~
t
,
i
x
2
+
M
~
t
,
i
x
+
R
~
t
,
i
,
i
f
x
+
η
¯
t
,
i
>
0and the optimal control is given by(85)
u
∗
=
-
G
t
,
i
′
-
1
ξ
¯
t
,
i
x
+
η
¯
t
,
i
,
i
f
x
+
η
¯
t
,
i
≤
0
,
0
,
i
f
x
+
η
¯
t
,
i
>
0
,where (86)
η
¯
t
,
i
=
M
^
t
,
i
R
^
t
,
i
.Proof.
Let(87)
Γ
1
i
=
t
,
x
,
i
∈
0
,
T
×
R
×
N
∣
x
+
η
¯
t
,
i
≤
0
,
Γ
2
i
=
t
,
x
,
i
∈
0
,
T
×
R
∣
x
+
η
¯
t
,
i
>
0
.
(i)
InΓ
1
i, V as given by (62) is well defined, with (88)
∂
V
i
t
,
x
∂
t
=
1
2
P
^
˙
t
,
i
x
2
+
M
^
˙
t
,
i
x
+
R
^
˙
t
,
i
,
∂
V
i
t
,
x
∂
x
=
P
^
t
,
i
x
+
M
^
t
,
i
,
∂
2
V
i
t
,
x
∂
x
2
=
P
^
t
,
i
.
Substituting them into the left-hand side (LHS) of (8), we obtain(89)
L
H
S
=
1
2
P
^
˙
t
,
i
+
P
^
t
,
i
A
t
+
1
2
∑
j
≠
i
q
i
j
P
^
t
,
i
-
P
^
t
,
j
x
2
M
^
˙
t
,
i
+
P
^
t
,
i
f
t
,
i
+
M
^
t
,
i
A
i
t
+
∑
j
≠
i
q
i
j
R
^
t
,
i
-
R
^
t
,
i
x
+
R
^
˙
t
,
i
+
M
^
t
,
i
f
t
,
i
+
∑
j
≠
i
q
i
j
R
^
t
,
i
-
R
^
t
,
i
inf
u
≥
0
⁡
1
2
u
′
G
t
,
i
′
G
t
,
i
u
+
B
t
,
i
x
+
M
^
t
,
i
P
^
t
,
i
u
.
Letη
¯
(
t
,
i
)
=
M
^
(
t
,
i
)
/
P
^
(
t
,
i
) and, by using Lemma A.1  α
¯
=
-
[
x
+
η
¯
(
t
,
i
)
]
>
0, it follows that the minimizer of (89) is achieved by(90)
u
∗
=
-
G
t
,
i
′
-
1
ξ
¯
t
,
i
x
+
η
¯
t
,
i
.
Substitutingu
∗
(
t
,
x
) back into (8) and noting (63)–(65), it immediately follows that L
H
S
=
0
.
Now, we will show thatV
¯ is a viscosity subsolution.
Letφ
i
∈
C
2
(
Q
) and choose (
t
¯
,
x
¯
)
∈
a
r
g
m
a
x
{
(
V
¯
i
-
φ
i
)
(
t
,
x
)
∣
(
t
,
x
)
∈
Q
¯
}
∩
Q; then, (91)
∂
φ
i
t
¯
,
x
¯
∂
t
=
∂
V
¯
i
t
¯
,
x
¯
∂
t
,
∂
φ
i
t
¯
,
x
¯
∂
x
=
∂
V
¯
i
t
¯
,
x
¯
∂
x
,
∂
2
V
¯
i
-
φ
i
∂
x
2
t
¯
,
x
¯
≤
0
⟹
P
¯
i
=
∂
2
V
¯
i
t
¯
,
x
¯
∂
x
2
≤
∂
2
φ
i
t
¯
,
x
¯
∂
x
2
and we obtain(92)
0
=
∂
t
V
¯
i
t
¯
,
x
¯
+
inf
u
≥
0
⁡
1
2
g
¯
i
t
¯
,
x
¯
,
u
g
¯
i
t
¯
,
x
¯
,
u
′
D
x
2
V
¯
i
t
¯
,
x
¯
+
f
¯
i
t
¯
,
x
¯
,
u
D
x
V
¯
i
t
¯
,
x
¯
-
∑
j
≠
i
q
i
j
V
¯
i
t
¯
,
x
¯
-
V
¯
j
t
¯
,
x
¯
≤
∂
t
φ
i
t
¯
,
x
¯
+
inf
u
≥
0
⁡
1
2
g
¯
i
t
¯
,
x
¯
,
u
g
¯
i
t
¯
,
x
¯
,
u
′
D
x
2
φ
i
t
¯
,
x
¯
+
f
¯
i
t
¯
,
x
¯
,
u
φ
i
t
¯
,
x
¯
-
∑
j
≠
i
q
i
j
V
¯
i
t
¯
,
x
¯
-
V
¯
j
t
¯
,
x
¯
.
Hence,V
¯
i is a viscosity subsolution.
(ii)
InΓ
2
i, we proceed similarly with (93)
∂
V
i
t
,
x
∂
t
=
1
2
P
~
˙
t
,
i
x
2
+
M
~
˙
t
,
i
x
+
R
~
˙
t
,
i
,
∂
V
i
t
,
x
∂
x
=
P
~
t
,
i
x
+
M
~
t
,
i
,
∂
2
V
i
t
,
x
∂
x
2
=
P
~
t
,
i
.
Substituting them into the left-hand side (LHS) of (8), we obtain(94)
L
H
S
=
1
2
P
~
˙
t
,
i
+
P
~
˙
t
,
i
A
t
,
i
+
1
2
∑
j
≠
i
q
i
j
P
~
˙
t
,
i
-
P
~
˙
t
,
j
x
2
M
~
˙
t
,
i
+
P
~
t
,
i
f
i
t
+
M
~
t
,
i
A
t
,
i
+
∑
j
≠
i
q
i
j
R
~
t
,
i
-
R
~
t
,
j
x
+
R
~
˙
t
,
i
+
M
~
˙
t
,
i
f
t
,
i
+
∑
j
≠
i
q
i
j
R
~
t
,
i
-
R
~
t
,
j
inf
u
≥
0
⁡
1
2
u
′
G
t
,
i
′
G
i
t
u
+
B
t
,
i
x
+
M
~
t
,
i
P
~
t
,
i
u
.
Sinceα
=
-
[
x
+
η
¯
(
t
,
i
)
]
>
0, the minimizer of (94) is (95)
u
i
∗
=
0
.
Substitutingu
∗ into (8), it is easy to show that V
~ satisfies HJBC equation (8) in Γ
2
i.
Now, we will show thatV
~ is a viscosity subsolution.
Letϕ
i
∈
C
2
(
Q
) and choose (
t
~
,
x
~
)
∈
argmin
⁡
{
(
V
~
i
-
φ
i
)
(
t
,
x
)
∣
(
t
,
x
)
∈
Q
¯
}
∩
Q; then, (96)
∂
ϕ
i
t
~
,
x
~
∂
t
=
∂
V
~
i
t
~
,
x
~
∂
t
,
∂
ϕ
i
t
~
,
x
~
∂
x
=
∂
V
~
i
t
~
,
x
~
∂
x
,
∂
2
V
~
i
-
ϕ
i
∂
x
2
t
~
,
x
~
≥
0
⟹
P
~
i
=
∂
2
V
~
i
t
~
,
x
~
∂
x
2
≥
∂
2
ϕ
i
t
~
,
x
~
∂
x
2
and we obtain(97)
0
=
∂
t
V
~
i
t
~
,
x
~
+
inf
u
≥
0
⁡
1
2
g
¯
i
t
~
,
x
~
,
u
g
¯
i
t
~
,
x
~
,
u
′
D
x
2
V
~
i
t
~
,
x
~
+
f
¯
i
t
~
,
x
~
,
u
D
x
V
~
i
t
~
,
x
~
-
∑
j
≠
i
q
i
j
V
~
i
t
~
,
x
~
-
V
~
j
t
~
,
x
~
≥
∂
t
ϕ
i
t
~
,
x
~
+
inf
u
≥
0
⁡
1
2
g
¯
i
t
~
,
x
~
,
u
g
¯
i
t
~
,
x
~
,
u
′
D
x
2
ϕ
i
t
~
,
x
~
+
f
¯
i
t
~
,
x
~
,
u
ϕ
i
t
~
,
x
~
-
∑
j
≠
i
q
i
j
V
~
i
t
~
,
x
~
-
V
~
j
t
~
,
x
~
.
Hence,V
~
i is a viscosity supersolution.We see that the value function V is a viscosity solution.
Remark  12. we see clearly that ∂
2
V
(
t
,
x
,
i
)
/
∂
x
2 does not exist in Q, since P
~
(
t
,
i
)
≠
P
^
(
t
,
i
). For this reason, we are required to work within the framework of viscosity solutions.


## 5. Efficient Strategies

Considerx
(
t
)
=
y
(
t
)
-
(
d
-
μ
). The problem A
(
μ
) is equivalent to the following problem: (98)
m
i
n
E
t
x
1
2
x
T
d
x
t
=
A
t
,
i
x
t
+
B
t
,
i
u
+
f
t
,
i
d
t
+
G
t
,
i
u
d
W
t
,
t
∈
s
,
T
x
0
=
y
0
-
d
-
μ
,where u
(
·
)
∈
L
F
2
(
0
,
T
;
R
+
m
) and(99)
A
t
,
i
=
r
t
,
i
,
B
t
,
i
=
b
1
t
,
i
-
r
t
,
i
,
…
,
b
n
t
,
i
-
r
t
,
i
,
f
t
,
i
=
d
-
μ
r
t
,
i
,
G
t
,
i
=
σ
i
1
t
,
i
,
…
,
σ
i
n
t
,
i
.Now, corresponding to (A.3), set (100)
π
¯
i
t
=
argmin
π
t
,
i
∈
0
,
∞
m
⁡
1
2
σ
t
,
i
-
1
π
t
,
i
+
σ
t
,
i
-
1
b
t
,
i
-
r
t
,
i
1
2
,
θ
¯
i
t
=
σ
i
t
-
1
π
t
,
i
+
σ
i
t
-
1
b
t
,
i
-
r
t
,
i
1
.


### 5.1. An Optimal Strategy

We present the optimal investment strategy for the problemA
(
μ
). The optimal control obtained in (85) translates into the following strategy: (101)
u
∗
≡
u
1
∗
,
…
,
u
m
∗
′
,
u
∗
=
-
σ
t
,
i
′
-
1
θ
¯
t
,
i
y
+
d
-
μ
+
η
¯
t
,
i
,
i
f
y
+
d
-
μ
+
η
¯
t
,
i
≤
0
,
0
,
i
f
y
+
d
-
μ
+
η
¯
t
,
i
>
0
.Theorem 13.
The optimal investment strategy to the problemA
(
μ
) is given by (101).


## 5.1. An Optimal Strategy

We present the optimal investment strategy for the problemA
(
μ
). The optimal control obtained in (85) translates into the following strategy: (101)
u
∗
≡
u
1
∗
,
…
,
u
m
∗
′
,
u
∗
=
-
σ
t
,
i
′
-
1
θ
¯
t
,
i
y
+
d
-
μ
+
η
¯
t
,
i
,
i
f
y
+
d
-
μ
+
η
¯
t
,
i
≤
0
,
0
,
i
f
y
+
d
-
μ
+
η
¯
t
,
i
>
0
.Theorem 13.
The optimal investment strategy to the problemA
(
μ
) is given by (101).


## 6. Efficient Frontier

Sincex
(
t
)
=
y
(
t
)
-
(
d
-
μ
), we obtain the solution of the original problem P
(
D
). Hence, for every fixed μ, we have(102)
m
i
n
u
·
∈
U
0
,
T
E
t
y
1
2
y
T
-
d
2
+
μ
E
t
y
y
T
-
d
=
min
u
·
∈
U
0
,
T
⁡
E
t
x
1
2
x
T
2
-
1
2
μ
2
=
V
i
0
0
,
x
-
1
2
μ
2
.Hence, the value function of P
(
D
) is given:(103)
V
i
0
0
,
x
-
1
2
μ
2
=
1
2
P
^
0
,
i
0
y
0
-
d
-
μ
2
+
M
^
0
,
i
0
y
0
-
d
-
μ
+
R
^
0
,
i
0
-
1
2
μ
2
1
y
+
d
-
μ
+
η
¯
t
,
i
≤
0
+
1
2
P
~
0
,
i
0
y
0
-
d
-
μ
2
+
M
~
0
,
i
0
y
0
-
d
-
μ
+
R
~
0
,
i
0
-
1
2
μ
2
1
y
+
d
-
μ
+
η
¯
t
,
i
>
0
.Note that the above value still depends on the Lagrange multiplier μ. To obtain the optimal value function, one needs to maximize the value of μ in (103).Proposition 14.
The efficient strategy of portfolio selection problem (50) corresponding to the expected terminal wealth E
t
y
y
(
T
)
=
d, as a function of time t and wealth y, is (104)
u
∗
=
-
σ
t
,
i
′
-
1
θ
¯
t
,
i
y
+
d
-
μ
∗
+
η
t
,
i
,
i
f
y
+
d
-
μ
∗
+
η
t
,
i
≤
0
,
0
,
i
f
y
+
d
-
μ
∗
+
η
t
,
i
>
0
.Moreover if (105)
μ
∗
=
argmax
μ
⁡
1
2
P
^
0
,
i
0
y
0
-
d
-
μ
2
+
M
^
0
,
i
0
y
0
-
d
-
μ
+
R
^
0
,
i
0
-
1
2
μ
2
1
y
+
d
-
μ
+
η
¯
t
,
i
≤
0
+
1
2
P
~
0
,
i
0
y
0
-
d
-
μ
2
+
M
~
0
,
i
0
y
0
-
d
-
μ
+
R
~
0
,
i
0
-
1
2
μ
2
1
y
+
d
-
μ
+
η
¯
t
,
i
>
0exists, the efficient frontier is given by (106)
V
a
r
y
T
=
V
0
,
x
,
i
0
-
1
2
μ
∗
2
.


## 7. Concluding Remarks

We analyzed mean-variance optimal portfolio selection for a market with regime switching. The formulation allows the market to have random switching with no-shorting constraint. Using techniques of stochastic linear quadratic control and the notion of viscosity solution, mean-variance efficient portfolio and efficient frontiers are derived explicitly in closed forms in terms of some systems of Riccati equation for which the solutions are provided by making use of the Magnus approach. The numerical application is in progress and it will be the subject of a new research paper.



---

*Source: 4543298-2016-05-04.xml*