Remark on optimal investment in a market with memory
We consider a financial market model driven by a Gaussian semimartingale with stationary increments. This driving noise process consists of n independent components and each component has memory described by two parameters. We extend results of the authors on optimal investment in this market.
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irk-123456789-44722009-11-20T12:00:24Z Remark on optimal investment in a market with memory Inoue, A. Nakano, Y. We consider a financial market model driven by a Gaussian semimartingale with stationary increments. This driving noise process consists of n independent components and each component has memory described by two parameters. We extend results of the authors on optimal investment in this market. 2007 Article Remark on optimal investment in a market with memory / A. Inoue, Y. Nakano // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 66-76. — Бібліогр.: 18 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4472 en Інститут математики НАН України |
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We consider a financial market model driven by a Gaussian semimartingale with stationary increments. This driving noise process
consists of n independent components and each component has memory described by two parameters. We extend results of the authors
on optimal investment in this market. |
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Inoue, A. Nakano, Y. |
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Inoue, A. Nakano, Y. Remark on optimal investment in a market with memory |
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Inoue, A. Nakano, Y. |
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Inoue, A. |
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Remark on optimal investment in a market with memory |
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Remark on optimal investment in a market with memory |
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Remark on optimal investment in a market with memory |
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Remark on optimal investment in a market with memory |
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Remark on optimal investment in a market with memory |
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remark on optimal investment in a market with memory |
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Інститут математики НАН України |
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2007 |
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Remark on optimal investment in a market with memory / A. Inoue, Y. Nakano // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 66-76. — Бібліогр.: 18 назв.— англ. |
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Theory of Stochastic Processes
Vol.13 (29), no.1-2, 2007, pp.66-76
AKIHIKO INOUE AND YUMIHARU NAKANO
REMARK ON OPTIMAL INVESTMENT IN A
MARKET WITH MEMORY
We consider a financial market model driven by a Gaussian semi-
martingale with stationary increments. This driving noise process
consists of n independent components and each component has mem-
ory described by two parameters. We extend results of the authors
on optimal investment in this market.
1. Introduction
In this paper, we extend results of Inoue and Nakano [12] on optimal
investment in a financial market model with memory. This market model
M consists of n risky and one riskless assets. The price of the riskless
asset is denoted by S0(t) and that of the ith risky asset by Si(t). We put
S(t) = (S1(t), . . . , Sn(t))
′, where A′ denotes the transpose of a matrix A.
The dynamics of the Rn-valued process S(t) are described by the stochastic
differential equation
dSi(t) = Si(t)
[
μi(t)dt +
∑n
j=1 σij(t)dYj(t)
]
, t ≥ 0, Si(0) = si,
i = 1, . . . , n,
(1)
while those of S0(t) by the ordinary differential equation
dS0(t) = r(t)S0(t)dt, t ≥ 0, S0(0) = 1,
where the coefficients r(t) ≥ 0, μi(t), and σij(t) are continuous deterministic
functions on [0,∞) and the initial prices si are positive constants. We
assume that the n× n volatility matrix σ(t) = (σij(t))1≤i,j≤n is nonsingular
for t ≥ 0.
Invited lecture.
2000 Mathematics Subject Classifications. Primary 91B28, 60G10; Secondary 62P05,
93E20.
Key words and phrases. Optimal investment, long term investment, processes with
memory, processes with stationary increments, Riccati equations.
66
OPTIMAL INVESTMENT IN A MARKET WITH MEMORY 67
We define the jth component Yj(t) of the Rn-valued driving noise process
Y (t) = (Y1(t), . . . , Yn(t))
′ of (1) by the autoregressive type equation
dYj(t)
dt
= −
∫ t
−∞
pje
−qj(t−s) dYj(s)
ds
ds +
dWj(t)
dt
, t ∈ R, Yj(0) = 0,
where W (t) = (W1(t), . . . , Wn(t))′, t ∈ R, is an Rn-valued standard Brown-
ian motion defined on a complete probability space (Ω,F , P ), the derivatives
dYj(t)/dt and dWj(t)/dt are in the random distribution sense, and pj ’s and
qj ’s are constants such that
0 < qj < ∞, −qj < pj < ∞, j = 1, . . . , n (2)
(see Anh–Inoue [1]). Equivalently, we may define Yj(t) by the moving-
average type representation
Yj(t) = Wj(t) −
∫ t
0
[∫ s
−∞
pje
−(qj+pj)(s−u)dWj(u)
]
ds, t ∈ R
(see [1], Examples 2.12 and 2.14). The components Yj(t), j = 1, . . . , n,
are Gaussian processes with stationary increments that are independent
of each other. Each Yj(t) has short memory that is described by the two
parameters pj and qj . Notice that, in the special case pj = 0, Yj(t) reduces
to the Brownian motion Wj(t).
The underlying information structure of the market model M is the
filtration (Ft)t≥0 defined by
Ft := σ (σ(Y (s) : 0 ≤ s ≤ t) ∪N ) , t ≥ 0,
where N is the P -null subsets of F . With respect to this filtration, Y (t) is a
semimartingale. In fact, we have the following two kinds of semimartingale
representations of Y (t) (see Anh et al. [2], Example 5.3, and Inoue et al.
[13], Theorem 2.1):
Yj(t) = Bj(t) −
∫ t
0
[∫ s
0
kj(s, u)dYj(u)
]
ds, t ≥ 0, j = 1, . . . , n, (3)
Yj(t) = Bj(t) −
∫ t
0
[∫ s
0
lj(s, u)dBj(u)
]
ds, t ≥ 0, j = 1, . . . , n, (4)
where, for j = 1, . . . , n, (Bj(t))t≥0 is the innovation process, i.e., an R-
valued standard Brownian motion such that
σ(Yj(s) : 0 ≤ s ≤ t) = σ(Bj(s) : 0 ≤ s ≤ t), t ≥ 0,
and Bj’s are independent of each other. The point of (3) and (4) is that
the deterministic kernels kj(t, s) and lj(t, s) are given explicitly by
kj(t, s) = pj(2qj + pj)
(2qj + pj)e
qjs − pje
−qjs
(2qj + pj)2eqjt − p2
je
−qjt
, 0 ≤ s ≤ t, (5)
lj(t, s) = e−(pj+qj)(t−s)lj(s), 0 ≤ s ≤ t, (6)
68 AKIHIKO INOUE AND YUMIHARU NAKANO
with
lj(s) := pj
[
1 − 2pjqj
(2qj + pj)2e2qjs − p2
j
]
, s ≥ 0. (7)
There already exist many references in which the standard driving noise,
that is, Brownian motion, is replaced by a different one, such as fractional
Brownian motion, so that the market model can capture memory effect. See,
e.g., Barndorff-Nielsen and Shephard [3], Hu et al. [11], Mishura [15] and
Heyde and Leonenko [10]. Among such models, the above model M driven
by Y (t) which is a Gaussian semimartingale with stationary increments is
possibly the simplest one. One advantage of M is that, assuming σij(t) =
σij , real constants, we can easily estimate the characteristic parameters pj,
qj and σij from stock price data. See [12], Appendix C, for this parameter
estimation from real market data.
In the market M, an agent with initial endowment x ∈ (0,∞) invests,
at each time t, πi(t)X
x,π(t) dollars in the ith risky asset for i = 1, . . . , n and
[1 − ∑n
i=1 πi(t)]X
x,π(t) dollars in the riskless asset, where Xx,π(t) denotes
the agent’s wealth at time t. The wealth process Xx,π(t) is governed by the
stochastic differential equation
dXx,π(t)
Xx,π(t)
=
[
1 −
∑n
i=1
πi(t)
] dS0(t)
S0(t)
+
∑n
i=1
πi(t)
dSi(t)
Si(t)
, Xx,π(0) = x.
Here, the self-financing strategy π(t) = (π1(t), . . . , πn(t))′ is chosen from the
admissible class
AT :=
{
π = (π(t))0≤t≤T :
π is an Rn-valued, progressively measurable
process satisfying
∫ T
0
‖π(t)‖2dt < ∞ a.s.
}
for the finite time horizon of length T ∈ (0,∞), where ‖ · ‖ denotes the
Euclidean norm of Rn. If the time horizon is infinite, π(t) is chosen from
A := {(π(t))t≥0 : (π(t))0≤t≤T ∈ AT for every T ∈ (0,∞)} .
Let α ∈ (−∞, 1) \ {0} and c ∈ R. In [12], the following three optimal
investment problems for the model M are considered:
V (T, α) := sup
π∈AT
1
α
E [(Xx,π(T ))α] , (8)
J(α) := sup
π∈A
lim sup
T→∞
1
αT
log E [(Xx,π(T ))α] , (9)
I(c) := sup
π∈A
lim sup
T→∞
1
T
log P
[
Xx,π(T ) ≥ ecT
]
. (10)
Problem (8) is the classical optimal investment problem that dates back to
Merton (cf. Karatzas and Shreve [14]). Hu et al. [11] studied this problem for
OPTIMAL INVESTMENT IN A MARKET WITH MEMORY 69
a Black–Scholes type model driven by fractional Brownian motion. Problem
(9) is a kind of long term optimal investment problem which is studied by
Bielecki and Pliska [4], and also by other authors under various settings,
including Fleming and Sheu [5,6], Nagai and Peng [16], Pham [17,18], Hata
and Iida [7], and Hata and Sekine [8,9]. Problem (10) is another type of
long term optimal investment problem, the aim of which is to maximize the
large deviation probability that the wealth grows at a higher rate than the
given benchmark c. Pham [17,18] studied this problem and established a
duality relation between Problems (9) and (10). Subsequently, this problem
is studied by Hata and Iida [7] and Hata and Sekine [8,9] under different
settings.
In [12], Problems (8)–(10) are studied for the market model M which
has memory. There, the following condition, rather than (2), is assumed in
solving (8)–(10):
0 < qj < ∞, 0 ≤ pj < ∞, j = 1, . . . , n. (11)
Thus, in [12], pj ≥ 0 rather than pj > −qj for j = 1, . . . , n. In this paper,
we focus on Problems (8) and (9), and extends the results of [12] so that
pj ’s may take negative values. The key to this extension is to show the
existence of solution for a relevant Riccati type equation.
In Sections 2 and 3, we review the results of [12] on Problems (8) and
(9), respectively, and, in Section 4, we extend these results.
2. Optimal investment over a finite horizon
In this section, we review the result of [12] on the finite horizon optimiza-
tion problem (8) for the market model M. We assume α ∈ (−∞, 1) \ {0}
and (11).
Let Y (t) = (Y1(t), . . . , Yn(t))
′ and B(t) = (B1(t), . . . , Bn(t))′ be the driv-
ing noise and innovation processes, respectively, described in the previous
section. We define an Rn-valued deterministic function λ(t) = (λ1(t), . . . ,
λn(t))′ by
λ(t) := σ−1(t) [μ(t) − r(t)1] , t ≥ 0, (12)
where 1 := (1, . . . , 1)′ ∈ Rn. For kj(t, s)’s in (5), we put
k(t, s) := diag(k1(t, s), . . . , kn(t, s)), 0 ≤ s ≤ t.
Let ξ(t) = (ξ1(t), . . . , ξn(t))
′ be the Rn-valued process
∫ t
0
k(t, s)dY (s), i.e.,
ξj(t) :=
∫ t
0
kj(t, s)dYj(s), t ≥ 0, j = 1, . . . , n. (13)
Let β be the conjugate exponent of α, i.e.,
(1/α) + (1/β) = 1.
70 AKIHIKO INOUE AND YUMIHARU NAKANO
Notice that 0 < β < 1 (resp. −∞ < β < 0) if −∞ < α < 0 (resp.
0 < α < 1).
We put l(t) := diag(l1(t), . . . , ln(t)), p := diag(p1, . . . , pn), and q :=
diag(q1, . . . , qn) with lj(t)’s as in (7). We also put
ρ(t) = (ρ1(t), . . . , ρn(t))′, b(t) = diag(b1(t), . . . , bn(t))
with
ρj(t) := −βlj(t)λj(t), t ≥ 0, j = 1, . . . , n, (14)
bj(t) := −(pj + qj) + βlj(t), t ≥ 0, j = 1, . . . , n. (15)
We consider the following one-dimensional backward Riccati equations: for
j = 1, . . . , n
Ṙj(t) − l2j (t)R
2
j (t) + 2bj(t)Rj(t) + β(1 − β) = 0, 0 ≤ t ≤ T,
Rj(T ) = 0.
(16)
We have the following result on the existence of solution to (16).
Lemma 1 ([12], Lemma 2.1). Let j ∈ {1, . . . , n}.
1. If pj = 0, then (16) has a unique solution Rj(t) ≡ Rj(t; T ).
2. If pj > 0 and −∞ < α < 0, then (16) has a unique nonnegative
solution Rj(t) ≡ Rj(t; T ).
3. If pj > 0 and 0 < α < 1, then (16) has a unique solution Rj(t) ≡
Rj(t; T ) such that Rj(t) ≥ bj(t)/l
2
j (t) for t ∈ [0, T ].
In what follows, we write Rj(t) ≡ Rj(t; T ) for the unique solution to
(16) in the sense of Lemma 1, and we put R(t) := diag(R1(t), . . . , Rn(t)).
For j = 1, . . . , n, let vj(t) ≡ vj(t; T ) be the solution to the following
one-dimensional linear equation:
v̇j(t) + [bj(t) − l2j (t)Rj(t; T )]vj(t) + β(1 − β)λj(t) − Rj(t; T )ρj(t) = 0,
0 ≤ t ≤ T, vj(T ) = 0.
(17)
We put v(t) ≡ v(t; T ) := (v1(t; T ), . . . , vn(t; T ))′.
For j = 1, . . . , n and (t, T ) ∈ Δ, write
gj(t; T ) := v2
j (t; T )l2j (t) + 2ρj(t)vj(t; T ) − l2j (t)Rj(t; T ) − β(1 − β)λ2
j(t),
where
Δ := {(t, T ) : 0 < T < ∞, 0 ≤ t ≤ T}.
Recall that we have assumed α ∈ (−∞, 1) \ {0} and (11). Here is the
solution to Problem (8) under the condition (11).
OPTIMAL INVESTMENT IN A MARKET WITH MEMORY 71
Theorem 2 ([12], Theorem 2.3). For T ∈ (0,∞), the strategy (π̂T (t))0≤t≤T
∈ AT defined by
π̂T (t) := (σ′)−1(t) [(1 − β)λ(t) − {1 − β + l(t)R(t;T )}ξ(t) + l(t)v(t;T )] (18)
is the unique optimal strategy for Problem (8). The value function V (T ) ≡
V (T, α) in (8) is given by
V (T ) =
1
α
[xS0(T )]α exp
[
(1 − α)
2
∑n
j=1
∫ T
0
gj(t; T )dt
]
.
3. Optimal investment over an infinite horizon
In this section, we review the result of [12] on the infinite horizon opti-
mization problem (9) for the financial market model M. Throughout this
section, we assume (11) and the following two conditions:
lim
T→∞
1
T
∫ T
0
r(t)dt = r̄ with r̄ ∈ R, (19)
lim
T→∞
λ(t) = λ̄ with λ̄ = (λ̄1, . . . , λ̄n)′ ∈ Rn. (20)
Here recall λ(t) = (λ1(t), . . . , λn(t))′ from (12).
Let α ∈ (−∞, 1) \ {0} and β be its conjugate exponent. Let j ∈
{1, . . . , n}. For bj(t) in (15), we have limt→∞ bj(t) = b̄j , where
b̄j := −(1 − β)pj − qj .
Notice that b̄j < 0. We consider the equation
p2
jx
2 − 2b̄jx − β(1 − β) = 0. (21)
When pj = 0, we write R̄j for the unique solution β(1 − β)/(2qj) of this
linear equation. If pj > 0, then
b̄2
j + β(1 − β)p2
j = (1 − β)[(pj + qj)
2 − q2
j ] + q2
j ≥ q2
j > 0,
so that we may write R̄j for the larger solution to the quadratic equation
(21). Let j ∈ {1, . . . , n}. For ρj(t) in (14), we have limt→∞ ρj(t) = ρ̄j, where
ρ̄j := −βpjλ̄j.
Define v̄j by (
b̄j − p2
j R̄j
)
v̄j + β(1 − β)λ̄j − R̄j ρ̄j = 0.
72 AKIHIKO INOUE AND YUMIHARU NAKANO
For j = 1, . . . , n and −∞ < α < 1, α = 0, we put
Fj(α) :=
(pj + qj)
2λ̄2
jα
[(1 − α)(pj + qj)2 + αpj(pj + 2qj)]
,
and
Gj(α) := (pj + qj) − qjα
−(1 − α)1/2 [(1 − α)(pj + qj)
2 + αpj(pj + 2qj)]
1/2
.
Recall ξ(t) from (13). Taking into account (18), we consider π̂ =
(π̂(t))t≥0 ∈ A defined by
π̂(t) := (σ′)−1(t)
[
(1 − β)λ(t) − (1 − β + pR̄)ξ(t) + pv̄
]
, t ≥ 0,
where R̄ := diag(R̄1, . . . , R̄n), v̄ := (v̄1, . . . , v̄n)′, and p := diag(p1, . . . , pn).
We define
α∗ := max(α1∗, . . . , αn∗) (22)
with
αj∗ :=
{
−∞ if −∞ < pj ≤ 2qj,
−3 − 8qj
pj−2qj
if 2qj < pj < ∞.
(23)
Notice that α∗ ∈ [−∞,−3).
Recall that we have assumed (11), (19) and (20). Here is the solution
to Problem (9) under the condition (11).
Theorem 3 ([12], Theorem 3.4). Let α∗ < α < 1, α = 0. Then π̂ is an
optimal strategy for Problem (9) with limit rather than limsup in (9). The
optimal growth rate J(α) in (9) is given by
J(α) = r̄ +
1
2α
n∑
j=1
Fj(α) +
1
2α
n∑
j=1
Gj(α).
4. Extensions
In this section, we extend Theorems 2 and 3 so that pj’s may take
negative values. The key is to extend Lemma 1 properly. We assume
α ∈ (−∞, 1) \ {0} and (2).
We put, for j = 1, . . . , n,
a1j(t) = lj(t)
2,
a2j(t) = βlj(t) − (pj + qj),
a3 = β(1 − β).
OPTIMAL INVESTMENT IN A MARKET WITH MEMORY 73
Then the Riccati equation (16) becomes
Ṙj(t) − a1j(t)R
2
j (t) + 2a2j(t)Rj(t) + a3 = 0, 0 ≤ t ≤ T,
Rj(T ) = 0.
Note that lj(t) is increasing and satisfies
lj(0) =
pj(pj + 2qj)
2(pj + qj)
≤ lj(t) ≤ pj , t ≥ 0.
Proposition 4. Let j ∈ {1, . . . , n}. We assume −qj < pj < 0 and
0 < α ≤
(
pj + qj
pj + qj − lj(0)
)2
. (24)
1. It holds that a2j(t) ≤ 0 for t ≥ 0.
2. It holds that a2j(t)
2 + a1j(t)a3 ≥ 0 for t ≥ 0.
Proof. We have
a2j(t) = βlj(t) − (pj + qj) ≤ βlj(0) − (pj + qj),
whence a2j(t) ≤ 0 if β ≥ (pj + qj)/lj(0) or
α ≤ (pj + qj)/lj(0)
[(pj + qj)/lj(0)] − 1
=
pj + qj
pj + qj − lj(0)
.
However, 0 < (pj + qj)/[pj + qj − lj(0)] < 1, whence
pj + qj
pj + qj − lj(0)
>
(
pj + qj
pj + qj − lj(0)
)2
.
Thus the first assertion follows.
We have
a2j(t)
2 + a1j(t)a3 = βlj(t)
2 − 2(pj + qj)lj(t)β + (pj + qj)
2
≥ βlj(0)2 − 2(pj + qj)lj(0)β + (pj + qj)
2
= β[{(pj + qj) − lj(0)}2 − (pj + qj)
2] + (pj + qj)
2,
whence a2j(t)
2 + a1j(t)a3 ≥ 0 if
β ≥ (pj + qj)
2
(pj + qj)2 − [(pj + qj) − lj(0)]2
.
However, this is equivalent to α ≤ [(pj + qj)/{pj + qj − lj(0)}]2. Thus the
second assertion follows.
Lemma 5. Let j ∈ {1, . . . , n}.
74 AKIHIKO INOUE AND YUMIHARU NAKANO
1. We assume −qj < pj < 0 and −∞ < α < 0. Then (16) has a unique
nonnegative solution Rj(t) ≡ Rj(t; T ).
2. We assume −qj < pj < 0 and (24). Then (16) has a unique solution
Rj(t) ≡ Rj(t; T ) such that Rj(t) ≥ R∗
j (t) for t ∈ [0, T ], where
R∗
j (t) :=
a2j(t) +
√
a2j(t)2 + a1j(t)a3
a1j(t)
.
Proof. The first assertion follows in the same way as in the proof of [12],
Lemma 2.1 (ii). Thus we prove the second assertion.
Notice that R∗
j (t) is the larger solution to the quadratic equation
a1j(t)x
2 − 2a2j(t)x − a3 = 0. Thus
a1j(t)R
∗
j (t)
2 − 2a2j(t)R
∗
j (t) − a3 = 0. (25)
Since a1j(t) > 0, a2j(t) ≤ 0 and a3 < 0, we see that R∗
j (t) ≤ 0. The equation
for V (t) := Rj(t) − R∗
j (t) becomes
V̇ (t) − a1j(t)V (t)2 + 2[a2j(t) − a1j(t)R
∗
j (t)]V (t) + Ṙ∗
j (t) = 0. (26)
By differenciating (25), we get
ȧ1j(t)R
∗
j (t)
2 + 2a1j(t)R
∗
j (t)Ṙ
∗
j (t) − 2ȧ2jR
∗
j (t) − 2a2j(t)Ṙ
∗
j (t) = 0,
whence
Ṙ∗
j (t) =
2ȧ2j(t)R
∗
j (t) − ȧ1j(t)R
∗(t)2
2
√
a2j(t)2 + a1j(t)a3
.
Now
2ȧ2j(t)R
∗
j (t) − ȧ1jR
∗
j (t)
2 = −2l̇j(t)R
∗
j (t){lj(t)R∗
j (t) − β}.
Since
a2j(t)
2 + a1j(t)a3 = βlj(t)
2 − 2(pj + qj)lj(t)β + (pj + qj)
2 < (pj + qj)
2,
we see that
lj(t)R
∗
j (t) − β =
1
lj(t)
[
−(pj + qj) +
√
a2j(t)2 + a1j(t)a3
]
> 0.
Thus Ṙ∗
j (t) ≥ 0. This and a1j(t) > 0 imply that (26) has a unique nonneg-
ative solution. The second assertion follows from this.
We define
α∗ := min(α∗
1, . . . , α
∗
n)
OPTIMAL INVESTMENT IN A MARKET WITH MEMORY 75
with
α∗
j :=
{ (
pj+qj
pj+qj−lj(0)
)2
if − qj < pj < 0,
1 if 0 ≤ pj < ∞.
Notice that α∗ ∈ (0, 1]. Recall α∗ from (22).
Taking the solution Rj(t) ≡ Rj(t; T ) of (16) in the sense of Lemma 1 or
5 and running through the same arguments as those in [12], Sections 2 and
3, we obtain the following extensions to Theorems 2 and 3.
Theorem 6. We assume (2) and −∞ < α < α∗, α = 0. Then the same
conclusions as those of Theorem 2 hold.
Theorem 7. We assume (2), (19), (20) and α∗ < α < α∗, α = 0. Then
the same conclusions as those of Theorem 3 hold.
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Department of Mathematics, Faculty of Science, Hokkaido Univer-
sity, Sapporo 060-0810, Japan.
E-mail address: inoue@math.sci.hokudai.ac.jp
Center for the Study of Finance and Insurance, Osaka University,
Toyonaka 560-8531, Japan.
E-mail address: y-nakano@sigmath.es.osaka-u.ac.jp
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