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Existence and stability of stationary solutions of
nonlinear diﬀerence equations under random
perturbations
Mei Zhu
∗
, Duo Wang and Maozheng Guo
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China
Abstract
Existence and stability of stationary solutions of nonlinear random diﬀerence equa
tions are studied in this note. Firstly, we give the weak conditions that guarantee
the stability of Lypanunov exponents un der small random perturbations. Secondly,
we ﬁnd out the conditions under which the ratio of the random norm and the stan
dard Euclidean norm has deterministic bounds. Based on these new results, we
provide easytouse conditions that guarantee the existence and almost sure stabil
ity of a stationary solution. In addition, we also prove that the stationary solution
converges with probability one to the ﬁxed point of the corresponding deterministic
system as the noise intensity tends to zero.
Keywords: diﬀerence equation; random perturbation; stationary process; Lya
punov exponents; random norm
AMS Subject C lassiﬁcation: 39A11; 91B02; 37N35; 37N40
∗
Corresponding author. Email: meizhu66@g mail.com
1
1 Introduction
Random diﬀerence equations have been studied in many contexts, particularly in mod
eling physical and economic systems subject to exogenous random perturbations. For the
random models, one of the basic problems is the longterm behavior of solutions, in partic
ular the existence and stability of stationary solutions of the systems. Theoretical study
in this direction is still far from maturity. Many complex practical models described by
nonlinear random diﬀerence equations have to be analyzed by numerical simulations (see
[4, 5, 8]).
Generally, r andom diﬀerence equations are tr eated by studying invariant measures or
stationary measures. Ara´ujo [1], Futia [7] and Zmarrou & Homburg [14] provided diﬀerent
conditions to guarantee the existence of invariant measures from diﬀerent perspectives.
Kifer [10] further studied the connections of properties of invariant measures in the ra ndom
case with those in the deterministic case.
Furthermore, the existence of stationary solutions for random diﬀerence equations
is also studied, which implies the existence of invariant measures. Arnold [2] gave the
stationary processes of hyperbolic aﬃne random dynamical systems. Tong [12] studied
the existence of stationary solutions for a special class of random diﬀerence equations
where the Jacobian matrices in the linear parts are deterministic. For more general
nonlinear ra ndom diﬀerence equations, Arnold & Boxler [3] provided a theorem that
additive noise turns a hyperbolic ﬁxed point into a stationary solution. However, the
conditions of the theorem ar e related to Lyapunov exponents and random norms, which
make it diﬃcult to apply directly theorem to practical models. Therefore, we try to
develop a new mathematical tool to build up a bridge between the theoretical study and
practical applications.
In this paper we provide easytouse conditions that guarantee the existence, stability
and convergence of stationary solutions. Firstly, we give weaker conditions to generate
the stability of Lyapunov exponents under small random perturbations by using some
ideas of [13]. In fact, Young [13] studied a more general case. However, some practical
models (see [15]) do not satisfy the conditions given by [13]. In order to deal with the spe
cial cases related to the practical models, we provide some weaker conditions than those
given by [13]. Consistent with [13], in this paper we also concentrate on twodimensional
random diﬀerence equations. Secondly, to our knowledge, for the ﬁrst time we give the
conditions to guarantee that the ratio of the random norm and the standard Euclidean
2
norm has deterministic bounds, which plays a crucial role in obtaining the theorem and
propositions of this paper. By using the relationship between the random norm and the
standard Euclidean norm and some ideas of [6], we ﬁnd out a deterministic neighbor
hood in which the Lipschitz constant in the case of random norm is deterministic and
suﬃciently small. Based on these new results and the idea of [3], we provide easytouse
conditions that g uar antee the existence of stationary solutions of nonlinear random dif
ference equations. In addition, we further obta in that the stationary solutions a re locally
asymptotically stable and converge to the corresponding ﬁxed po ints of the deterministic
system as the noise intensity tends to zero both in the almost sure sense, which implies
that the stationary measures also converge weakly to the Dirac measure supported on
the corresponding deterministic ﬁxed point as the intensity of noise tends to zero. These
properties are signiﬁcant for studies of practical models.
In order to obtain the above results, here we make two basic assumptions. On the one
hand we assume that the perturbations are bounded. This assumption is motiva t ed by
some pra ctical backgrounds, for example, dividends in ﬁnancial markets var y randomly
and in a bounded range. See also [14] for reasons o f considering bounded noise. On the
other hand we assume that the random map is C
2
not only with respect to state variables
but also with respect t o random perturbations. In fact, many pra ctical models are all
analytic with respect to all va r ia bles and parameters as well as random perturbations.
Araujo [1] a nd Zmarrou & Homburg [14] have the similar assumptions.
The rest of this paper is set out as follows. Section 2 introduces the preliminaries and
the main theorem. Section 3 provides three lemmas. Section 4 proves the main theorem.
Section 5 presents several properties of stationary solutions. Section 6 gives a conclusion.
2 Random diﬀerence equation
Now consider nonlinear diﬀerence equation
Y
n
= A(ǫε
n
)Y
n−1
+ H(ǫε
n
, Y
n−1
) + h(ǫε
n
), n ∈ Z, (2.1)
where Y
n
∈ R
2
, A(·) ∈ C
1
([−b, b], Gl(2, R)), H(·, ·) ∈ C
2
([−b, b] × R
2
, R
2
), h(·) ∈ C
2
([−b, b], R
2
). Here b is a positive number. ǫε
n
denotes the random perturbation, where
ǫ(∈ [0, 1]) is a constant revealing the corresponding perturbation intensity. As ǫ = 0,
the diﬀerence equation (2.1) is deterministic. Suppose that {ε
n
} is independent and
ident ically distributed (i.i.d.) with noise ε
n
independent of {Y
m
, m < n}. Further, let
3
ν denote the probability measure of ε
n
on [−b, b]. Let Ω := [−b, b]
Z
be the biinﬁnite
product of the interval [−b, b] with itself. For each ω ∈ Ω, let its n’th coordinate (ω)
n
equal some value of ε
n
. That is, ω = (· · · ε
−1
, ε
0
, ε
1
· · · ). Further, let σ : Ω → Ω be the
shift op erator, i.e., (σω)
n
= (ω)
n+1
.
For convenience of expression, set
Y
n
= A(ǫε
n
)Y
n−1
+ H(ǫε
n
, Y
n−1
) + h(ǫε
n
)
:= A
ǫ
(ε
n
)Y
n−1
+ H
ǫ
(ε
n
, Y
n−1
) + h
ǫ
(ε
n
). (2.2)
Furthermore, we assume
A
ǫ
(ε
n
) = A(ǫε
n
) =
a
1
(ǫε
n
) a
2
(ǫε
n
)
a
3
(ǫε
n
) a
4
(ǫε
n
)
:=
a
1
ǫ
(ε
n
) a
2
ǫ
(ε
n
)
a
3
ǫ
(ε
n
) a
4
ǫ
(ε
n
)
.
Deﬁne the linear cocycle
Φ
ǫ
(n, ω) :=
A
ǫ
(ε
n
) · · · A
ǫ
(ε
2
)A
ǫ
(ε
1
), for n ≥ 1,
id, for n = 0,
where id denotes a 2 × 2 identity matrix. When ǫ = 0, A
0
(ε
n
) = A(0) is a constant
matrix independent of noise. Let A
0
:=
a
1
0
a
2
0
a
3
0
a
4
0
denote A
0
(ε
n
) a nd Φ
0
(n) denote
Φ
0
(n, ω) = A
n
0
.
Let u ∈ P
1
denote the direction represented by the unit vector u ∈ R
2
, where P
1
is
the projective space on R
2
. For suﬃciently small ǫ and t ∈ [−b, b], deﬁne a map A
ǫ
(t)(u)
: P
1
→ P
1
by A
ǫ
(t)(u) = A
ǫ
(t)u. Then the map A
ǫ
(ε
n
) induces a Markov chain on P
1
.
Further, let E be a Borel subset of P
1
. The transition probabilities ar e deﬁned by
P
ǫ
(u, E) = ν{t ∈ [−b, b] : A
ǫ
(t)(u) ∈ E}.
In the following proof, we also need the deﬁnition and some properties of the random
norm. First of all, we give the deﬁnition of the random norm. Here we use a deﬁnition
similar to t hat in [3].
Assume that Lyapunov exponents {λ
ǫ
i
} of Φ
ǫ
satisfy λ
ǫ
2
≤ λ
ǫ
1
< 0. Set λ
ǫ
s
:= λ
ǫ
1
. Fix a
κ(ǫ) > 0 such that λ
ǫ
s
+ κ(ǫ) < 0. For any Y : Ω → R
2
, let
kY (ω)k
ω
:=
∞
X
i=0
e
−(λ
ǫ
s
+κ(ǫ))i
kΦ
ǫ
(i, ω)Y (ω)k (2.3)
4
denote the random norm of Y w.r.t the cocycle Φ
ǫ
(n, ω), where k · k is the standard
Euclidean norm in R
2
. The crucial advantage of this random norm is that for all Y : Ω →
R
2
kΦ
ǫ
(n, ω)Y (ω)k
σ
n
ω
≤ e
(λ
ǫ
s
+κ)n
kY (ω)k
ω
, ∀ n = 0 , 1, 2, · · · . (2.4)
This is because, for n = 0, 1, 2, · · ·
kΦ
ǫ
(n, ω)Y (ω)k
σ
n
ω
≤ e
(λ
ǫ
s
+κ)n
∞
X
i=−n
e
−(λ
ǫ
s
+κ)(i+n)
kΦ
ǫ
(i + n, ω)Y (ω)k = e
(λ
ǫ
s
+κ)n
kY (ω)k
ω
.
Further, deﬁne the random norm for the linear operator Φ
ǫ
(n, ω) by
kΦ
ǫ
(n, ω)k
ω ,σ
n
ω
= sup
kY k
ω
6=0
kΦ
ǫ
(n, ω)Y (ω)k
σ
n
ω
kY (ω)k
ω
.
Hence based on (2.4),
kΦ
ǫ
(n, ω)k
ω ,σ
n
ω
≤ e
(λ
ǫ
s
+κ)n
.
Note that the above random norms for vectors and matrices as well as κ all depend on
the noise intensity ǫ.
Set kY k := ess sup kY (ω)k
ω
.
The main result of this paper is to provide a theorem that guarantees the existence of
stationary solutions to the nonlinear random diﬀerence equation (2.1) , i.e. the following
Theorem 2.1. Its proof will be given in Section 4.
Theorem 2.1 Fo r random diﬀerence equation (2.1), assume
(i) The two eigenva l ues of A(0) are both n onzero and less than one in modulus,
(ii) For an y given u ∈ P
1
, the probability measure P
ǫ
(u, ·) in duced by A
ǫ
(ε
1
)(u) is abso
lutely continuous with respect to Lebesgue measure m(·) on P
1
,
(iii) H(ǫε
n
, 0) = 0, D
Y
H(ǫε
n
, 0) = 0 for any ǫ ∈ [0, 1] and any n ∈ Z,
(iv) h(0) = 0.
Then for suﬃciently small ǫ, there exists ex a ctly one stationary solution in the neighbor
hood of the origin whose initial value ξ
ǫ
(ω) satisﬁes
kξ
ǫ
k ≤ Kǫ, (2.5)
where K > 0 i s a constant i ndependent of ǫ.
5
Remark 2.1 Conditions (i), (iii) and (iv) imply that the origin 0 is a stable ﬁxed point
of the deterministic diﬀerence equation Y
n
= A(0)Y
n−1
+ H(0, Y
n−1
) + h(0). In fact, any
random diﬀerence equation Y
n
= F (ǫε
n
, Y
n−1
) can be tra nsformed into equation (2.1),
where F (·, ·) ∈ C
2
([−b, b] × R
2
, R
2
), if its correspo nding deterministic diﬀerence equation
Y
n
= F (0, Y
n−1
) has a ﬁxed point.
Remark 2.2 If the measure ν is absolutely continuous with respect to Leb esgue measure
on [−b, b] and a rctan
a
3
ǫ
(t) cos θ+ a
4
ǫ
(t) sin θ
a
1
ǫ
(t) cos θ+ a
2
ǫ
(t) sin θ
is monotone or piecewise monotone with respect to
t for any given θ ∈ [−
π
2
,
π
2
], then the condition (ii) in Theorem 2.1 holds.
Remark 2.3 Since ǫ = 0 is a degenerate case where 0 is a trivial solution, in the following
we consider the case that ǫ > 0 only.
3 Several lemmas
In order to prove Theorem 2.1, at ﬁrst we provide several lemmas.
Lemma 3.1 Let A(·) : [−b, b] → Gl(2, R) be C
1
and A(0) be invertible. Assume that the
transition probability P
ǫ
(u, ·) is absolutely continuous with respect to Lebesgue measure
m(·) on P
1
for any given u ∈ P
1
. Then the Lyapunov exponents λ
ǫ
i
→ λ
0
i
as ǫ → 0,
i = 1, 2.
Proof. In the case that λ
0
1
> λ
0
2
, there are two unit eigenvectors {u
i
} corresponding
to {λ
0
i
} respectively for the matrix A
0
= A(0). Let θ
i
denote the angle corresponding to
the vector u
i
. Since P
1
∼
=
S
1
/{θ, θ + π}, then we can identify P
1
with S
1
/{θ, θ + π} and
use θ instead of ¯u. Further, let U (S
1
/{θ, θ + π}) denote the space of Borel probability
measures on S
1
/{θ, θ + π}. Deﬁne the operator L
ǫ
on U (S
1
/{θ, θ + π}) by
L
ǫ
(·)(E) =
Z
P
1
P
ǫ
(θ, E) d(·).
Then a probability measure µ on S
1
/{θ, θ + π} is an invariant measure if and only if for
any Borel subset E in S
1
/{θ, θ + π}
L
ǫ
(µ)(E) = µ(E).
Invariant measures on compact space S
1
/{θ, θ + π} always exist and are absolutely con
tinuous with respect to Lebesgue measure, because we assume that P
ǫ
(θ, ·) is absolutely
continuous with respect to Lebesgue measure for any θ.
6
Following t he similar argument as in [13], the maximal Lypunov exponent λ
ǫ
1
can be
expressed by
λ
ǫ
1
=
Z
P
1
×[−b,b]
log kA(t
′
)uk d(µ
ǫ
× ν
ǫ
),
where µ
ǫ
is any invariant measure on P
1
and ν
ǫ
is the measure induced by t
′
= ǫt on
[−ǫb, ǫb]. Hence in order to prove the lemma it is suﬃcient to prove that the invariant
measure µ
ǫ
converges weakly to δ
u
1
as ǫ → 0.
If we deﬁne a sequence of probability measures from any probability measure µ by
1
n
n−1
X
k=0
(L
k
ǫ
µ), ∀ µ ∈ U (S
1
/{θ, θ + π}), n = 1, 2, 3, · · · ,
then any (weak) limit of this sequence is an invariant measure, reversely any invariant
measure can be constructed in this way. See for example, [2].
Thus to show the lemma, it suﬃces to show that there is a small neighborhood N of
θ
2
in S
1
/{θ, θ + π} such that µ
ǫ
(N ) → 0 as ǫ → 0, i.e.,
lim
ǫ→0
lim
n→∞
1
n
n−1
X
k=0
(L
k
ǫ
µ)(N ) = 0, (3.1)
for some µ ∈ U (S
1
/{θ, θ + π}).
Since A
0
is an invertible matrix, then (a
1
0
cos θ + a
2
0
sin θ)
2
+ (a
3
0
cos θ + a
4
0
sin θ)
2
6= 0
for any θ ∈ S
1
. Thus based on
A
0
cos θ
sin θ
=
a
1
0
cos θ + a
2
0
sin θ
a
3
0
cos θ + a
4
0
sin θ
,
A
0
(θ) = arctan
a
3
0
cos θ + a
4
0
sin θ
a
1
0
cos θ + a
2
0
sin θ
or
π
2
, ∀θ ∈ S
1
/{θ, θ + π}.
Note that in the space S
1
/{θ, θ + π}, θ and π + θ are identical.
Hence
A
0
′
(θ) =
a
1
0
a
4
0
− a
2
0
a
3
0
(a
1
0
cos θ + a
2
0
sin θ)
2
+ (a
3
0
cos θ + a
4
0
sin θ)
2
.
If θ = θ
2
, then
a
1
0
cos θ
2
+ a
2
0
sin θ
2
= ±e
λ
0
2
cos θ
2
, a
3
0
cos θ
2
+ a
4
0
sin θ
2
= ±e
λ
0
2
sin θ
2
.
Hence
A
0
′
(θ
2
) =
±e
λ
0
1
e
λ
0
2
e
2λ
0
2
cos
2
θ
2
+ e
2λ
0
2
sin
2
θ
2
= ±e
λ
0
1
−λ
0
2
.
7
Thus we can choose a positive constant δ
∗
1
(<
1
4
θ
2
− θ
1
) such that
A
0
′
(θ) ≥ e
λ
0
1
−λ
0
2
−ǫ
0
> 1, ∀θ ∈ [θ
2
− δ
∗
1
, θ
2
+ δ
∗
1
], (3.2)
where ǫ
0
is a positive number satisfying ǫ
0
<
λ
0
1
−λ
0
2
2
. Take N = (θ
2
− δ
∗
1
, θ
2
+ δ
∗
1
). Let
ρ
ǫ
(θ) denote the density of P
ǫ
(θ, ·) for given θ. In the following we ﬁrst show that supp
ρ
ǫ
(θ
2
± δ
∗
1
) is disjoint from the δ
∗
1
neighborhood of θ
2
for suﬃciently small ǫ. Namely the
case 2 in the proof of Theorem 1 in [13] never occurs under our situation.
If the two eigenvalues of A
0
are both p ositive, then A
0
′
(θ
2
) = e
λ
0
1
−λ
0
2
> 1 . Thus based
on A
0
(θ
2
) = θ
2
, for the above suﬃciently small positive constant δ
∗
1
,
A
0
(θ
2
− δ
∗
1
) − θ
2
< −δ
∗
1
, A
0
(θ
2
+ δ
∗
1
) − θ
2
> δ
∗
1
.
That is,
θ
∗
−
:= A
0
(θ
2
− δ
∗
1
) < θ
2
− δ
∗
1
:= θ
−
, θ
∗
+
:= A
0
(θ
2
+ δ
∗
1
) > θ
2
+ δ
∗
1
:= θ
+
.
Further, since A(t
′
) is continuous with respect to t
′
and A
0
is invertible, then ∃ǫ
1
∈ (0, 1],
such that as ǫ < ǫ
1
,
a
1
ǫ
(t)a
4
ǫ
(t) − a
2
ǫ
(t)a
3
ǫ
(t) 6= 0, ∀ t ∈ [−b, b].
Thus A(ǫt)(θ) is well deﬁned. Moreover, for θ = θ
−
, θ
+
, θ
1
, there exists ǫ
2
∈ (0, ǫ
1
] such
that for any ǫ < ǫ
2
and t ∈ [−b, b],
A(ǫt)(θ
−
) − A
0
(θ
−
) <
1
4
(θ
−
− θ
∗
−
),
A(ǫt)(θ
+
) − A
0
(θ
+
) <
1
4
(θ
∗
+
− θ
+
),
A(ǫt)(θ
1
) − A
0
(θ
1
) < δ
∗
1
.
Note that A
0
(θ
1
) = θ
1
. Therefore, supp ρ
ǫ
(θ
2
± δ
∗
1
) and supp ρ
ǫ
(θ
1
) are disjoint from
the δ
∗
1
neighborhood of θ
2
as ǫ < ǫ
2
, respectively. If the two eigenvalues of A
0
are both
negative or have diﬀerent signs, the same result still holds following the similar argument.
Let µ be any Borel probability measure on S
1
/{θ, θ+π} which is absolutely continuous
with respect to Lebesgue measure. Let η
0
denote the density of µ on S
1
/{θ, θ + π}, and
let η
i
denote the density of L
i
ǫ
µ on S
1
/{θ, θ + π}, i = 1, 2, · · ·. Note that the densities η
0
and η
1
have the relation
η
1
(θ) =
Z
[−b,b]
η
0
A
ǫ
(t)
−1
(θ)
·
A
ǫ
(t)
−1
′
(θ)
dν(t), (3.3)
8
where
A
ǫ
(t)
−1
(θ) is the inverse mapping of A
ǫ
(t)(θ) with respect to θ for given t. In
the case that ǫ < ǫ
2
, since supp ρ
ǫ
(θ
2
± δ
∗
1
) and supp ρ
ǫ
(θ
1
) are disjoint from the δ
∗
1

neighborhood of θ
2
and A
ǫ
(t)(θ) is strictly monotone with respect to θ for any g iven t,
then for any θ 6∈ [θ
2
−δ
∗
1
, θ
2
+δ
∗
1
], A
ǫ
(t)(θ) 6∈ [θ
2
−δ
∗
1
, θ
2
+δ
∗
1
]. That is, for θ ∈ [θ
2
−δ
∗
1
, θ
2
+δ
∗
1
],
(A
ǫ
(t))
−1
(θ) ∈ [θ
2
− δ
∗
1
, θ
2
+ δ
∗
1
].
Furthermore, for any ǫ < ǫ
2
and t ∈ [−b, b], based on
∂A
ǫ
(t)(θ)
∂θ
6= 0, (A(t
′
))
−1
(θ) is continu
ously diﬀerentiable with respect to t
′
:= ǫt and θ. Thus ((A(t
′
))
−1
)
′
(θ) is continuous with
respect to t
′
and θ. Hence there exists ǫ
3
(0 < ǫ
3
< ǫ
2
), such that as 0 < ǫ < ǫ
3
, for any
t ∈ [−b, b],
max
θ∈N
((A(ǫt))
−1
)
′
(θ) ≤ e
ǫ
0
max
θ∈N
((A
0
)
−1
)
′
(θ) ≤ e
−λ
0
1
+λ
0
2
+2ǫ
0
, (3.4)
where the second inequality is based on (3.2).
Let eη
i
:= max{η
i
(θ) : θ − θ
2
 ≤ δ
∗
1
}, i = 0, 1, · · · . Without loss of generality, we
assume eη
0
< ∞ and then eη
i
is well deﬁned for i ≥ 1. Hence based on (3.3) and (3.4),
eη
1
≤ eη
0
· e
−λ
0
1
+λ
0
2
+2ǫ
0
,
where 0 < e
−λ
0
1
+λ
0
2
+2ǫ
0
< e
−λ
0
1
+λ
0
2
+λ
0
1
−λ
0
2
= 1 .
Recursively,
eη
k
≤ eη
0
· e
k(−λ
0
1
+λ
0
2
+2ǫ
0
)
, k = 1, 2, · · · .
Therefore,
1
n
n−1
X
k=0
(L
k
ǫ
µ)N ≤
2δ
∗
1
n
n−1
X
k=0
eη
k
≤
2δ
∗
1
n
eη
0
1
1 − e
−λ
0
1
+λ
0
2
+2ǫ
0
→ 0, as n → ∞,
where the last convergence is uniform with respect to ǫ ∈ (0, ǫ
3
). That is to say that we
have the equation (3.1). Hence this lemma holds in the case that λ
0
1
> λ
0
2
.
The proof for the case that λ
0
1
= λ
0
2
is easy, and hence we omit it here.
Lemma 3.2 If all the assumptions of Lemm a 3.1 are satisﬁed and λ
0
2
≤ λ
0
1
< 0, then for
suﬃciently small ǫ, the random norm and the standard norm have the f ollowing relation
kY (ω)k ≤ kY (ω)k
ω
≤ BkY (ω)k, ∀ω ∈ Ω, (3.5)
where B is a positive con s tant independent of ǫ.
9
Proof. At ﬁrst, according to the deﬁnition of the random norm (2.3),
kY (ω)k ≤ kY (ω)k
ω
. (3.6)
On the other hand, since the Lypunov exponents λ
0
2
≤ λ
0
1
< 0, then the sp ectral radius
of the matrix A
0
denoted by ρ(A
0
) satisﬁes ρ(A
0
) < 1. Hence there is a norm k · k
a
in
R
2
and a corresponding operator norm k · k
a
for matrices such that kA
0
k
a
< 1, see for
example, [11]. Let 0 < α < β be such t hat
αkY k ≤ kY k
a
≤ βkY k, ∀Y ∈ R
2
.
Moreover, based on the assumptions, ∃ t
′
ǫ
∈ [−ǫb, ǫb] such that for any t ∈ [−b, b]
kA
ǫ
(t)k
a
= k A(ǫt)k
a
= kA(t
′
)k
a
≤ max
t
′
∈[−ǫb,ǫb]
kA(t
′
)k
a
= kA(t
′
ǫ
)k
a
= e
M
ǫ
,
where M
ǫ
= log kA(t
′
ǫ
)k
a
. In particular, for ǫ = 0, M
0
= log kA
0
k
a
< 0. Thus, for any
i = 1, 2, · · · ,
kΦ
ǫ
(i, ω)k
a
≤ kA
ǫ
(ε
i
)k
a
· · · k A
ǫ
(ε
2
)k
a
kA
ǫ
(ε
1
)k
a
≤ e
iM
ǫ
, ∀ ω ∈ Ω.
Hence
kY (ω)k
ω
≤
∞
X
i=0
e
−i(λ
ǫ
s
+κ(ǫ))
kΦ
ǫ
(i, ω)Y (ω)k
≤
1
α
∞
X
i=0
e
−i(λ
ǫ
s
+κ(ǫ))
kΦ
ǫ
(i, ω)Y (ω)k
a
≤
β
α
∞
X
i=0
e
i(M
ǫ
−λ
ǫ
s
−κ(ǫ))
kY (ω)k.
Furthermore, since M
ǫ
→ M
0
and λ
ǫ
s
→ λ
0
s
as ǫ → 0, then ∃ ǫ
4
∈ (0, ǫ
3
] such that for any
ǫ < ǫ
4
, we have
λ
ǫ
s
− λ
0
s
 < −
1
4
M
0
, (M
ǫ
− λ
ǫ
s
) − (M
0
− λ
0
s
) < −
1
4
M
0
.
Take κ ≡ κ(ǫ) =
1
2
M
0
− λ
0
s
(> M
0
− λ
0
s
≥ 0). Then
λ
ǫ
s
+ κ(ǫ) < λ
0
s
−
1
4
M
0
+
1
2
M
0
− λ
0
s
=
1
4
M
0
< 0,
M
ǫ
− λ
ǫ
s
− κ(ǫ) < M
0
− λ
0
s
−
1
4
M
0
−
1
2
M
0
+ λ
0
s
=
1
4
M
0
< 0 .
Thus as ǫ < ǫ
4
,
kY (ω)k
ω
≤
β
α
∞
X
i=0
e
i(M
ǫ
−λ
ǫ
s
−κ(ǫ))
kY (ω)k ≤
β
α
∞
X
i=0
e
iM
0
4
kY (ω)k = BkY (ω)k, (3.7)
10
where B =
β
α
∞
P
i=0
e
iM
0
4
=
β
α
·
1
1−e
M
0
4
.
Therefore, for suﬃciently small ǫ, combination of (3.6) with (3.7) gives
kY (ω)k ≤ kY (ω)k
ω
≤ BkY (ω)k.
Note that in the following κ =
1
2
M
0
− λ
0
s
. Further, we set Λ = 1 − e
M
0
4
(∈ (0, 1)).
Lemma 3.3 Under the conditions of Lemma 3. 2, let H : [−b, b] × R
2
→ R
2
be a C
2
function with H
ǫ
(ε
1
, 0) = H(ǫε
1
, 0) = 0 and D
Y
H
ǫ
(ε
1
, 0) = D
Y
H(ǫε
1
, 0) = 0 for any
ǫ ∈ [0, 1]. Then for suﬃciently small ǫ, there exists a cons tant r ∈ (0, 1] independent of ǫ
and ω such that for any kY
i
(ω)k ≤ r ≤ 1, i = 1, 2
kH
ǫ
(ε
1
, Y
1
(ω)) − H
ǫ
(ε
1
, Y
2
(ω))k
σω
≤ LkY
1
(ω) − Y
2
(ω)k
ω
, ∀ω ∈ Ω,
where L =
Λ
2
> 0.
Proof. Since H(·, ·) is a C
2
function, for any kY
1
(ω)k ≤ 1, kY
2
(ω)k ≤ 1, we have
kD
Y
H(ǫε
1
, Y
1
(ω)) − D
Y
H(ǫε
1
, Y
2
(ω))k ≤ JkY
1
(ω) − Y
2
(ω)k < ∞, ∀ǫ ∈ [0, 1], ε
1
∈ [−b, b],
where J is a positive constant independent of ǫ and ω as well as Y
i
, i = 1, 2 . Take r ∈ (0, 1].
Since D
Y
H(ǫε
1
, , 0) = 0, then for any kY (ω)k ≤ r,
kD
Y
H(ǫε
1
, Y (ω))k = kD
Y
H(ǫε
1
, Y (ω)) − D
Y
H(ǫε
1
, , 0)k ≤ JkY (ω)k ≤ Jr.
Hence for any kY
i
(ω)k ≤ r ≤ 1, i = 1, 2,
kH
ǫ
(ε
1
, Y
1
(ω)) − H
ǫ
(ε
1
, Y
2
(ω))k ≤ sup
kY (ω)k≤r
kD
Y
H(ǫε
1
, Y (ω))kkY
1
(ω) − Y
2
(ω)k
≤ JrkY
1
(ω) − Y
2
(ω)k. (3.8)
From Lemma 3.2, as ǫ < ǫ
4
, then
kH
ǫ
(ε
1
, Y
1
(ω)) − H
ǫ
(ε
1
, Y
2
(ω))k
σω
≤ BkH
ǫ
(ε
1
, Y
1
(ω)) − H
ǫ
(ε
1
, Y
2
(ω))k
≤ BJrkY
1
(ω) − Y
2
(ω)k
≤ BJrkY
1
(ω) − Y
2
(ω)k
ω
.
Set
r = min{
L
BJ
, 1} = min{
Λ
2BJ
, 1}.
Therefore, for any kY
i
(ω)k ≤ r, i = 1, 2, as ǫ < ǫ
4
, we have
kH
ǫ
(ε
1
, Y
1
(ω)) − H
ǫ
(ε
1
, Y
2
(ω))k
σω
≤ BJrkY
1
(ω) − Y
2
(ω)k
ω
≤ LkY
1
(ω) − Y
2
(ω)k
ω
.
11
4 Proof of Theorem 2. 1
Proof. We ﬁrst consider a special case and then provide the proof in a general case.
Step 1: If H ≡ 0, then (2.1) is an aﬃne random diﬀerence equation. In this case, for
suﬃciently small ǫ, the var ia t io n of constant s formula gives the solution with its initial
value
ξ
ǫ
(ω) :=
0
X
i=−∞
Φ
ǫ
(−i, σ
i
ω)h
ǫ
(ε
i
). (4.1)
The expression makes sense, for
kξ
ǫ
k = ess sup kξ
ǫ
(ω)k
ω
≤ ess sup
0
X
i=−∞
kΦ
ǫ
(−i, σ
i
ω)k
σ
i
ω ,ω
kh
ǫ
(ε
i
)k
σ
i
ω
≤ ess sup
0
X
i=−∞
e
−i(λ
ǫ
s
+κ)
kh
ǫ
(ε
i
)k
σ
i
ω
≤
1
1 − e
λ
ǫ
s
+κ
kh
ǫ
k
≤
1
Λ
kh
ǫ
k, (4.2)
where the last inequality holds because that 0 < 1 − e
1
4
M
0
= Λ < 1 − e
λ
ǫ
s
+κ
as ǫ < ǫ
4
.
Note that all ess sup kh
ǫ
(ε
i
)k
σ
i
ω
, i ∈ Z are equal, we denote them by kh
ǫ
k.
Based on the assumptions that h(·) ∈ C
2
and h(0) = 0, ∃ǫ
5
∈ (0, ǫ
4
] such that as
ǫ < ǫ
5
,
kh
ǫ
(ε
0
)k = kh(ǫε
0
)k ≤
C
0
b
ǫε
0
 ≤ C
0
ǫ,
where C
0
is a positive constant independent of ǫ.
Furthermore,
kh
ǫ
(ε
0
)k
ω
= k h(ǫε
0
)k
ω
≤ Bkh(ǫε
0
)k ≤ BC
0
ǫ. (4.3)
Hence
kξ
ǫ
k ≤
kh
ǫ
k
Λ
≤
BC
0
ǫ
Λ
< ∞.
On the other hand, according to the cocycle property,
A
ǫ
(ε
1
)ξ
ǫ
(ω) + h
ǫ
(ε
1
) =
1
X
i=−∞
Φ
ǫ
(1 − i, σ
i
ω)h
ǫ
(ε
i
) = ξ
ǫ
(σω).
12
Hence t he stochastic process with initial value ξ
ǫ
(ω) is a stationary process. Namely,
{ξ
ǫ
n
(ω)} := {ξ
ǫ
(σ
n
ω)} is a stationary solution to (2.1).
Furthermore, we will show that the stationary solution is unique in the neighborhood
of the orig in. If ζ
ǫ
1
, ζ
ǫ
2
are both stationary solutions of (2.1), then ζ
ǫ
:= ζ
ǫ
1
− ζ
ǫ
2
is also one
stationary solution and satisﬁes ζ
ǫ
n+1
= A
ǫ
(ε
n+1
)ζ
ǫ
n
. However, based on Lemma 3.1 and
Lemma 3.2, the linearized system ζ
ǫ
n+1
= A
ǫ
(ε
n+1
)ζ
ǫ
n
is hyperbolic. Hence ζ
ǫ
≡ 0. That
is, the stationary solution is unique.
Step 2: Consider now the full equation (2.1).
Just as [3], we now deﬁne a similar space having similar properties. Let
L
∞
:= {Y : Ω → R
2
measurable with kY k < ∞}.
Then L
∞
is a Banach space. Thus for r > 0 in Lemma 3.3,
D
r
:= {Y ∈ L
∞
: kY k ≤ r}.
is also complete. In the following we will prove the existence of a stationary solution
using the Banach ﬁxed point theorem. Hence we ﬁrst deﬁne an operator and then test
the conditions o f the theorem.
(1) We now deﬁne a nonlinear operator T
ǫ
: D
r
→ L
∞
by
T
ǫ
Y (ω) =
0
X
i=−∞
Φ
ǫ
(−i, σ
i
ω)
e
h
ǫ
(σ
i
ω), (4.4)
where
e
h
ǫ
(ω) = H
ǫ
(ε
0
, Y (σ
−1
ω)) + h
ǫ
(ε
0
).
The deﬁnition makes sense since, according to Lemma 3.3 and (4.3),
k
e
h
ǫ
(ω)k
ω
≤ kH
ǫ
(ε
0
, Y (σ
−1
ω))k
ω
+ kh
ǫ
(ε
0
)k
ω
≤ Lr + BC
0
ǫ.
That is,
k
e
h
ǫ
k = ess sup k
e
h
ǫ
(ω)k
ω
≤ Lr + BC
0
ǫ. (4.5)
Furthermore, the comparison of (4.1) with (4.4) shows that T
ǫ
Y is just the initial value
of the unique stationary solution of the aﬃne equation Y
n
= A
ǫ
(ε
n
)Y
n−1
+
e
h
ǫ
(ε
n
). Hence
from ( 4.2),
kT
ǫ
Y k ≤
1
1 − e
λ
ǫ
s
+κ
k
e
h
ǫ
k ≤
1
Λ
k
e
h
ǫ
k. (4.6)
13
(2) T
ǫ
maps D
r
to D
r
.
At ﬁrst, ∃ ǫ
6
=
rΛ
2BC
0
> 0, such that as 0 < ǫ < ǫ
6
, Lr + BC
0
ǫ ≤ rΛ. Furthermore,
based on (4.5) and (4.6), as 0 < ǫ < ǫ
7
:= min{ǫ
5
, ǫ
6
}, we have
kT
ǫ
Y k ≤
Lr + BC
0
ǫ
Λ
≤
rΛ
Λ
= r.
(3) T
ǫ
is contracting on D
r
.
From Lemma 3.3 and the property of the random norm (2.4), fo r any Y
1
, Y
2
∈ D
r
, we
have
kT
ǫ
Y
1
(ω) − T
ǫ
Y
2
(ω)k
ω
= k
0
X
i=−∞
Φ
ǫ
(−i, σ
i
ω)(H
ǫ
(ε
i
, Y
1
(σ
i−1
ω)) − H
ǫ
(ε
i
, Y
2
(σ
i−1
ω)))k
ω
≤
0
X
i=−∞
kΦ
ǫ
(−i, σ
i
ω)k
σ
i
ω ,ω
kH
ǫ
(ε
i
, Y
1
(σ
i−1
ω)) − H
ǫ
(ε
i
, Y
2
(σ
i−1
ω))k
σ
i
ω
≤
0
X
i=−∞
e
−i(λ
ǫ
s
+κ)
LkY
1
(σ
i−1
ω) − Y
2
(σ
i−1
ω)k
σ
i−1
ω
≤
1
2
kY
1
− Y
2
k.
That is,
kT
ǫ
Y
1
− T
ǫ
Y
2
k = ess sup kT
ǫ
Y
1
(ω) − T
ǫ
Y
2
(ω)k
ω
≤
1
2
kY
1
− Y
2
k.
Therefore, by the Banach ﬁxed point theorem, (2.1) has a stationary solution whose
initial value ξ
ǫ
(ω) satisﬁes kξ
ǫ
(ω)k ≤ kξ
ǫ
(ω)k
ω
≤ kξ
ǫ
k ≤ r for almost all ω (and is unique
with this property).
Furthermore, for the stationa ry solution with initial value ξ
ǫ
(ω), based on Lemma 3.3
and (4.6),
kξ
ǫ
(ω)k
ω
= kT
ǫ
ξ
ǫ
(ω)k
ω
≤ kT
ǫ
ξ
ǫ
k
≤
1
Λ
kH
ǫ
(ε
0
, ξ
ǫ
(σ
−1
ω)) + h
ǫ
(ε
0
)k
≤
1
Λ
(kH
ǫ
(ε
0
, ξ
ǫ
(σ
−1
ω)) − H
ǫ
(ε
0
, 0)k + kh
ǫ
k)
≤
1
Λ
(Lkξ
ǫ
k + k h
ǫ
k).
That is,
kξ
ǫ
k = ess sup kξ
ǫ
(ω)k
ω
≤
1
Λ
(Lkξ
ǫ
k + kh
ǫ
k).
14
Hence for almost all ω,
kξ
ǫ
(ω)k ≤ kξ
ǫ
(ω)k
ω
≤ kξ
ǫ
k ≤
kh
ǫ
k
Λ − L
=
2kh
ǫ
k
Λ
.
Moreover based on (4.3 ) , for almost all ω,
kξ
ǫ
(ω)k ≤ kξ
ǫ
k ≤
2kh
ǫ
k
Λ
≤
2BC
0
ǫ
Λ
. (4.7)
Take K =
2BC
0
Λ
. Then the proof of Theorem 2.1 is complete.
5 Properties
Proposition 5.1 The stationary solution obtained in Theorem 2.1 is measurable.
Proof. We start with ξ
ǫ,0
≡ 0 (which is obviously measurable) and recursively deﬁne
ξ
ǫ,k+1
(ω) := T
ǫ
ξ
ǫ,k
(ω), k = 0, 1, 2 · · · .
Based on the assumptions of theorem 2.1, A(·) is C
1
, H(·, ·), h(·) are C
2
, and hence they
are all measurable. Thus according to the deﬁnition of T
ǫ
in (4.4), ξ
ǫ,1
(ω) := T
ǫ
ξ
ǫ,0
(ω) is
measurable. Hence recursively ξ
ǫ,k
(ω) is measurable. In addition, the proof of (2) of step
2 in section 4 also shows that ξ
ǫ,k
(ω) ∈ D
r
.
By the Banach ﬁxed point theorem on D
r
,
lim
k→∞
ξ
ǫ,k
= ξ
ǫ
.
That is, for any eε > 0, ∃K(eε) > 0 such that ∀k ≥ K(eε),
kξ
ǫ,k
− ξ
ǫ
k < eε.
Furthermore, due to the relation of norms, as k > K(eε),
kξ
ǫ,k
(ω) − ξ
ǫ
(ω)k ≤ kξ
ǫ,k
(ω) − ξ
ǫ
(ω)k
ω
≤ kξ
ǫ,k
− ξ
ǫ
k < eε.
Hence for almost all ω,
lim
k→∞
ξ
ǫ,k
(ω) = ξ
ǫ
(ω).
Given that the ﬁxed point ξ
ǫ
(ω) is a pointwise limit of measurable functions, then it is
measurable.
Note that the proof follows Lemma 7.3.4 in [2].
15
Proposition 5.2 As ǫ → 0, the s tationa ry solution obtained in Theorem 2.1 converges
almost surely to zero, which is an isola ted ﬁxed poin t in the deterministic case.
Proof. Since according to the assumptions of Theorem 2.1, h
0
= h(0) = 0 and H
0
(ε
n
, 0) =
H(0, 0) = 0, then 0 is a ﬁxed point for the deterministic diﬀerence equation, namely the
equation ( 2.1) with ǫ = 0. Based on (4.7), fo r almost all ω,
kξ
ǫ
(ω)k ≤ kξ
ǫ
(ω)k
ω
≤
2BC
0
ǫ
Λ
→ 0, as ǫ → 0.
That is, the random ﬁxed point converges almost surely to the deterministic ﬁxed point
as the noise intensity tends to zero.
Note that in general study of stochastic stability, only weak convergence can be derived.
Proposition 5.3 The stationary solution obtained in Theorem 2.1 is locally exponentially
stable with probability one.
Proof. If ζ
n
(ω) is any other solution of (2.1) on Z
+
with initial va lue kζ
0
k ≤ r, then
ζ
n+1
= A
ǫ
(ε
n+1
)ζ
n
+ H
ǫ
(ε
n+1
, ζ
n
) + h
ǫ
(ε
n+1
). (5.1)
Since kζ
0
(ω)k
ω
≤ r, based on the above equation (5.1) and 1−e
λ
ǫ
s
+κ
= 1−e
λ
ǫ
s
+
1
2
M
0
−λ
0
s
> Λ,
as ǫ < ǫ
7
,
kζ
1
k
σω
≤ kA
ǫ
(ε
1
)k
ω ,σ
1
ω
kζ
0
k
ω
+ kH
ǫ
(ε
1
, ζ
0
)k
σω
+ kh
ǫ
(ε
1
)k
σω
≤ re
λ
ǫ
s
+κ
+ Lr + BC
0
ǫ ≤ r.
We recursively obtain kζ
n
k
σ
n
ω
≤ r. Hence
kζ
n
k ≤ r.
On the other hand, ξ
ǫ
n
is the stationary solution of model (2.1) with initial value ξ
ǫ
0
(ω) =
ξ
ǫ
(ω). That is,
ξ
ǫ
(σ
n+1
ω) := ξ
ǫ
n+1
= A
ǫ
(ε
n+1
)ξ
ǫ
n
+ H
ǫ
(ε
n+1
, ξ
ǫ
n
) + h
ǫ
(ε
n+1
). (5.2)
Subtracting (5.2) from (5.1) yields the equation
ζ
n+1
− ξ
ǫ
n+1
= A
ǫ
(ε
n+1
)(ζ
n
− ξ
ǫ
n
) + H
ǫ
(ε
n+1
, ζ
n
) − H
ǫ
(ε
n+1
, ξ
ǫ
n
).
Set ζ
∗
n+1
:= ζ
n+1
− ξ
ǫ
n+1
, n = 0, 1, 2, · · · . Then ζ
∗
n+1
satisﬁes the following equation:
ζ
∗
n+1
= A
ǫ
(ε
n+1
)ζ
∗
n
+ H
ǫ
(ε
n+1
, ζ
∗
n
+ ξ
ǫ
n
) − H
ǫ
(ε
n+1
, ξ
ǫ
n
). (5.3)
16
Since kζ
∗
n
+ ξ
ǫ
n
k
σ
n
ω
= kζ
n
k
σ
n
ω
≤ kζ
n
k ≤ r and kξ
ǫ
n
k
σ
n
ω
≤ kξ
ǫ
k ≤ r, then kH
ǫ
(ε
n+1
, ζ
∗
n
+
ξ
ǫ
n
) − H
ǫ
(ε
n+1
, ξ
ǫ
n
)k
σ
n+1
ω
≤ Lkζ
∗
n
k
σ
n
ω
, based on Lemma 3.3.
Thus
kζ
∗
n+1
k
σ
n+1
ω
≤ (e
λ
ǫ
s
+κ
+ L)kζ
∗
n
k
σ
n
ω
≤ (e
λ
ǫ
s
+κ
+ L)
n+1
kζ
∗
0
k
ω
= γ
n+1
kζ
∗
0
k
ω
,
where γ = e
λ
ǫ
s
+κ
+ L ≤ 1 − Λ +
Λ
2
= 1 −
Λ
2
∈ (0, 1).
For any eε > 0, without loss of generality, we assume eε < r ≤ 1. Take
e
δ =
eε
B
. Since B > 1,
e
δ < 1. As kζ
0
− ξ
ǫ
0
k <
e
δ < 1, for almost all ω and any n = 0, 1, 2, · · ·
kζ
n
− ξ
ǫ
n
k ≤ kζ
n
− ξ
ǫ
n
k
σ
n
ω
≤ γ
n
kζ
0
− ξ
ǫ
0
k
ω
< B
e
δ = eε.
Hence the stationary solution with the initial value ξ
ǫ
(ω) is locally stable. Furthermore,
for almost all ω
kζ
n
− ξ
ǫ
n
k ≤ kζ
n
− ξ
ǫ
n
k
σ
n
ω
< γ
n
kζ
0
− ξ
ǫ
0
k
ω
< γ
n
eε < γ
n
, for any n ∈ N.
Therefore, the stationary solution with the initial value ξ
ǫ
(ω) is locally exponentially
stable with pro bability one.
6 Conclus i on and future direc tions
In this paper we provide some new easytouse conditions that guara ntee the existence
and stability of nonlinear random diﬀerence equations with a view of application. First,
we obtain the stability of Lyapunov exponents for a special case under weaker conditions,
namely Lemma 3 .1 . Second, we derive a new r elationship between the random norm a nd
the standard norm, i.e., the ratio of the random norm and the standard norm has deter
ministic bounds, namely Lemma 3.2. However, in general theory of random dynamical
systems, the bound of the ratio of the random norm a nd the standard norm is random
depending on sample paths (see [2]). This result plays a crucial role in obtaining a de
terministic neighborhood in which the Lipschitz constant in the case of random norm
is deterministic and suﬃciently small, namely Lemma 3 .3 . But in the general theory o f
random dynamical systems, the neighborhoo d is random depending on sample paths (see
[2, 6]). Eventually, by integrat ing the lemmas we provide a theorem that guar antees the
existence of stationary solutions under easytouse conditions, as stated in Theorem 2.1.
In addition, we also obtain several prop erties including measurability, local exponent sta
bility with probability one and the relationship between the stationary solution and the
17
corresponding deterministic ﬁxed point. Under new conditions, the general result of local
asymptotical stability in probability given by [2] is improved. Moreover, the result that
the stationary solution converges to the deterministic ﬁxed point with probability o ne as
the noise intensity tends to zero also implies that the stationary measure converges to the
Dirac measure suppo r t ed on the deterministic ﬁxed point with the noise intensity tending
to zero.
That is, for nonlinear random diﬀerence equations, we not only establish the easyto
use conditions to guarantee the existence of stationary solutions but also study stability
and relation o f the stationary solution and its corresponding deterministic ﬁxed point.
These properties are imp ortant for practical models. For example, in [9] the original
random model is studied by using an approximate deterministic model which is given
by setting the random dividend in the random model as a constant, its mean. With
the theorem and propositions in this paper, it can be proved that there exists a unique
stable stationary solution in a small neighborhood of the ﬁxed point of the corresponding
deterministic diﬀerence equation provided the perturbation intensity is suﬃciently small
(see [15 ]). Thus from the dynamic point of view, the dynamic prop erties are very close for
the random model and the corresponding deterministic model near the ﬁxed point since
random dividends in real markets are generally small. That is, the method they used in [9]
is meaningful at least for the dynamics of the model near the ﬁxed point. Hence in some
sense this paper builds up a bridge between some theoretical researches of mathematics
and their applications to practical models.
In future research, ﬁrstly, it is possible to provide new easytouse conditions that
guarantee the existence and stability stationary solutions of high dimensional nonlinear
random diﬀerence equations with a view of application. Secondly, it is signiﬁcant to
generalize the i.i.d. noise to more general cases. Finally, it is also interesting to study
various kinds of bifurcations based on the obtained results fr om a dynamical point of
view.
Acknowledgement s
Financial support f r om the NSF of China (No. 10571003, 10871005) is acknowledged.
18
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