SCIENCE CHINA
Mathematics
. ARTICLES .
June 2013 Vol. 56 No. 6: 1169–1180
doi: 10.1007/s11425-012-4553-1
Further results on existence-uniqueness for stochastic
functional differential equations
XU DaoYi1, WANG XiaoHu1 & YANG ZhiGuo2,∗
2College
1Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China;
of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China
Email: [email protected], [email protected], [email protected]
Received May 19, 2011; accepted July 23, 2012; published online February 19, 2013
Abstract The aim of this paper is to develop some basic theories of stochastic functional differential equations
(SFDEs) under the local Lipschitz condition in continuous functions space C. Firstly, we establish a global
existence-uniqueness lemma for the SFDEs under the global Lipschitz condition in C without the linear growth
condition. Then, under the local Lipschitz condition in C, we show that the non-continuable solution of SFDEs
still exists if the drift coefficient and diffusion coefficient are square-integrable with respect to t when the state
variable equals zero. And the solution of the considered equation must either explode at the end of the maximum
existing interval or exist globally. Furthermore, some more general sufficient conditions for the global existenceuniqueness are obtained. Our conditions obtained in this paper are much weaker than some existing results. For
example, we need neither the linear growth condition nor the continuous condition on the time t. Two examples
are provided to show the effectiveness of the theoretical results.
Keywords
stochastic functional differential equations, existence, uniqueness
MSC(2010)
34K50, 34A12
Citation: Xu D Y, Wang X H, Yang Z G. Further results on existence-uniqueness for stochastic functional differential
equations. Sci China Math, 2013, 56: 1169–1180, doi: 10.1007/s11425-012-4553-1
1
Introduction
Stochastic differential equations (SDEs) play a very important role in formulation and analysis in mechanical, electrical, control engineering and physical sciences, economic and social sciences. Therefore,
the theory of SDEs has been developed very quickly [1–4, 6–8, 12, 13, 15, 16]. Recently, the investigation for stochastic functional differential equations (SFDEs) has attracted the considerable attention
of researchers and many qualitative theories of SFDEs have been obtained. Many important results
can be found in [9–11, 14, 17–19] and references cited therein. The important representative works
on the existence-uniqueness have been discussed under the Lipschitz conditions in two spaces. One
is L2 (Ω; C), for example, Mohammed [11] gave the global existence-uniqueness theorem under the global
Lipschitz condition and the linear growth condition in L2 (Ω; C); Xu et al. [19], Ning and Liu [9] developed
basic theories of SFDEs and delay stochastic evolution equations under the local Lipschitz condition in
L2 (Ω; C), respectively. The other is C or Rn [3, 6, 10, 12]. These results in L2 (Ω; C) are perfect in theory.
But it is inconvenient to check the local Lipschitz condition in L2 (Ω; C). So, most of existing results are
established under the Lipschitz condition in C or Rn .
∗ Corresponding
author
c Science China Press and Springer-Verlag Berlin Heidelberg 2013
math.scichina.com
www.springerlink.com
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Friedman [3] considered the following SDE of Itô-type,
dx(t) = B(t, x(t))dt + σ(t, x(t))dω(t),
t ∈ [0, T ],
(1.1)
with the initial condition x(0) = x0 , where T > 0 is a constant. By employing the local Lipschitz
condition in Rn and the global linear growth condition, Friedman gave the global existence-uniqueness
theorem of SDE (1.1) (see [3, Theorem 2.2] and the earlier works can be found in [1, 2, 4, 6]).
In [10], Mao generalized the above result in [3] to the following SFDE:
dx(t) = B(t, xt )dt + σ(t, xt )dω(t),
t0 t < T,
(1.2)
where x ∈ Rn is the state variable, B(t, xt ) is drift coefficient and σ(t, xt ) diffusion coefficient, and the
initial condition
xt0 = ξ = {ξ(θ) : −τ θ 0} is an Ft0 -measurable
C-valued random variable such that Eξ2 < ∞.
(1.3)
However, the linear growth condition is somewhat restrictive and many SFDEs do not obey it. By
Δ
employing the stopping time δn of the sample paths on every cylinder [0, ∞) × Uk (Uk = {x : |x| k}, k
is a positive constant), Khasminskii [7] only required that the Lipschitz condition and linear growth
condition held in the cylinder [0, ∞) × Uk . That is, for each k = 1, 2, . . . , there are constants Kk and
ck > 0 such that
|B(t, x) − B(t, x̂)| ∨ |σ(t, x) − σ(t, x̂)| Kk |x − x̂|,
∀ t ∈ [0, ∞),
(1.4)
for all x, x̂ ∈ Rn with |x| ∨ |x̂| k, and
|B(t, x)| ∨ |σ(t, x)| ck (1 + |x|),
∀ t ∈ [0, ∞),
(1.5)
for all x ∈ Rn with |x| k.
In [14], the above result in [7] has been generalized to the SFDEs while the locally linear growth
condition is replaced by a weaker one
sup (|B(t, 0)| ∨ |σ(t, 0)|) < ∞.
(A)
0t<∞
But, for all of the above results in [7, 14], we need to check the other condition
P (δ = ∞) = 1,
δ = lim δn is a stopping time,
n→∞
(B)
by a Lyapunov function.
Although the above results can be applied to many nonlinear SFDEs as demonstrated in [7,14], condition (A) on B and σ is restrictive, and it is in fact not easy to check the condition (B). The results will
become much more powerful without the conditions (A) and (B).
The main aim of this paper is to give some results without the conditions (A) and (B) of that sort
results. This paper is organized as follows. We firstly obtain a global existence-uniqueness lemma for
SFDEs under the global Lipschitz condition in C without the linear growth condition. Then, under the
local Lipschitz condition in C, we show that the non-continuable solution of SFDEs given by Xu et al. [19]
still exists if the drift coefficient and diffusion coefficient are square-integrable with respect to t when the
state variable equals zero. And the solution of the considered equation must either explode at the end
of the maximum existing interval or exist globally. Furthermore, some more general sufficient conditions
for the global existence-uniqueness are obtained. Our new theorems give better results while conditions
imposed are much weaker than some existing results. For example, we need neither the linear growth
condition nor the continuous condition on the time t. Therefore, our new results cannot only be applied
much more easily but also cover a much wider class of SFDEs. In the last section, two examples are
provided to show the effectiveness of the theoretical results.
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Preliminaries
In this section, we introduce some notations and recall some basic definitions.
n
Let | · | be any norm in Rn . Denote by Rn+ the nonnegative cone in Rn , that is, Rn+ = {x
∈ R : xi 0, ∀ i = 1, . . . , n}. If A is a matrix or vector, denote by |A| its trace norm, i.e., |A| = trace(AT A).
Δ
Let C = C([−τ, 0]; Rn ) denote the family of all continuous functions ϕ from [−τ, 0] to Rn with the norm
ϕ = sup−τ s0 |ϕ(s)|, where τ > 0.
Let (Ω, F , {Ft }tt0 , P ) be a complete probability space and ω(t) = (ω1 (t), . . . , ωm (t))T be an mdimensional Brownian motion defined on (Ω, F , {Ft }tt0 , P ). Mp ([a, b]; Rn×m ) = {f : f is n × m-matrixb
valued-measurable Ft -adapted process and E a |f (t)|p dt < ∞} for p > 0. We denote M2 ([a, b]; Rn ) =
M2 ([a, b]; Rn×1 ). Especially, we let N p ([a, b]; Rn×m ) = {f : f is n × m-matrix-valued-measurable and
b
p
p
n
p
n×1
). Note that the functions in N 2 ([a, b]; Rn×m ) are
a |f (t)| dt < ∞}, and N ([a, b]; R ) = N ([a, b]; R
2
n×m
2
n×m
) ⊂ M ([a, b]; R
). Denote by L2 (Ω; C) the family of all C-valued
deterministic and N ([a, b]; R
random variables ξ with Eξ2 < ∞. A stochastic process x(t) (t ∈ [a, b]) is said to be continuous if x(t)
is continuous on [a, b] almost surely.
Let C1,2 (R × Rn ; R+ ) denote the family of all nonnegative functions V (t, x) on R × Rn which are twice
continuously differentiable in x and once in t. For each V (t, x) ∈ C1,2 (R × Rn ; R+ ), we define an operator
LV , associated with the SFDE (1.2), from R × Rn to R by
1
LV (t, x) = Vt (t, x) + Vx (t, x)B(t, xt ) + trace[σ T (t, xt )Vxx σ(t, xt )],
2
2
∂V (t, x)
∂V (t, x)
∂V (t, x)
∂ V (t, x)
, Vx (t, x) =
, Vxx (t, x) =
,...,
.
Vt (t, x) =
∂t
∂x1
∂xn
∂xi ∂xj n×n
In the following, we always assume that B : C × [t0 , T ) → Rn and σ : C × [t0 , T ) → Rn×m are both
Borel measurable.
Definition 2.1. Let J = [t0 − τ, a) or J = [t0 − τ, a] with a ∈ [t0 , T ). Rn -value stochastic process x(t)
defined on J is called a solution of (1.2) and (1.3) if x(t) is continuous, Ft -adapted and satisfies xt0 = ξ
and
t
x(t) = ξ(0) +
t
B(s, xs )ds +
t0
σ(s, xs )dω(s),
∀ t ∈ [t0 , a) or [t0 , a]
t0
almost surely. The solution x(t) of (1.2) and (1.3) on interval J is said to be unique if any other solution
x̄(t) on interval J is indistinguishable from it, that is,
P {x(t) = x̄(t) for all t ∈ J} = 1.
Definition 2.2.
small enough,
The solution x(t, ω) of (1.2) and (1.3) is said to explode at t̄ > t0 if for any ε > 0
P
sup |x(t)| = ∞ > 0.
|t−t̄|<ε
Definition 2.3. Let F : [t0 , T ) × C → Rn . The functional F (t, φ) is said to satisfy the local Lipschitz
condition in φ on [t0 , b] ⊂ [t0 , T ) if for any n > 0 there is a constant Kn = Kn (b) > 0 such that
|F (t, ϕ) − F (t, ψ)| Kn ϕ − ψ,
(2.1)
for all t ∈ [t0 , b] and those ϕ, ψ ∈ C with ϕ ∨ ψ n. Moreover, F is said to satisfy the local Lipschitz
condition in φ in [t0 , T ) if (2.1) holds for any b ∈ (t0 , T ).
3
Main results
The following lemma is a generalization of the classical results in [3, Theorem 5.1.1] and [10,
Theorem 5.2.2] since it does not need the linear growth condition.
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Assume that for any b ∈ [t0 , T ),
B(t, 0) ∈ N 2 ([t0 , b]; Rn ) and σ(t, 0) ∈ N 2 ([t0 , b]; Rn×m ).
(3.1)
If B and σ satisfy the global Lipschitz condition in [t0 , b] × C, that is, there is a constant L > 0 such that
for all ϕ, ψ ∈ C and t ∈ [t0 , b],
|B(t, ϕ) − B(t, ψ)|2 ∨ |σ(t, ϕ) − σ(t, ψ)|2 Lϕ − ψ2 ,
(3.2)
then the initial problem (1.2) and (1.3) has a unique continuous solution x(t) on [t0 − τ, b].
Proof.
Since the solution of (1.2) with (1.3) satisfies the following integral equation:
t
t
B(s, xs )ds +
σ(s, xs )dω(s), ∀ t ∈ [t0 , b],
x(t) = ξ(0) +
t0
(3.3)
t0
we may let xnt0 = ξ, n = 0, 1, . . . , and define Picard sequence, x0 (t) = ξ(0) for t ∈ [t0 , b] and
t
t
xn (t) = ξ(0) +
B(s, xn−1
)ds
+
σ(s, xn−1
)dω(s), t ∈ [t0 , b], n = 1, 2, . . . ,
s
s
t0
(3.4)
t0
by the standard methods of dealing with the SDEs (see [3, 13]).
From (3.2),
|B(t, xt )|2 2|B(t, 0)|2 + 2|B(t, xt ) − B(t, 0)|2 2|B(t, 0)|2 + 2Lxt 2 ,
(3.5)
|σ(t, xt )| 2|σ(t, 0)| + 2|σ(t, xt ) − σ(t, 0)| 2|σ(t, 0)| + 2Lxt .
(3.6)
2
2
2
2
2
Next, we will prove that, for n = 1, 2, . . . ,
xn (t) is Ft -measurable and continuous, t ∈ [t0 , b],
C [M̄ (t − t )]n−1
0
0
, t ∈ [t0 , b],
E sup |xn (s) − xn−1 (s)|2 (n − 1)!
t0 st
E
sup |xn (s)|2 C0 (1 + Eξ2 )eC0 (t−t0 ) , t ∈ [t0 , b],
t0 −τ st
(3.7)
(3.8)
(3.9)
where
M̄ = 2L(b − t0 + 4),
Δ
C0 = 4 + 3M̄ + 2(b − t0 ) M̄ Eξ2 + 3
b
b
|B(s, 0)|2 ds + 24
|σ(s, 0)|2 ds.
t0
(3.10)
t0
From (1.3), (3.1), (3.5) and (3.6), one obtains that
B(s, x0s ) ∈ M2 ([t0 , b]; Rn ) and σ(s, x0s ) ∈ M2 ([t0 , b]; Rn×m ).
(3.11)
Then, from Theorem 4.3.2 and its remark in [3], x1 (t) defined by x0 (t) and (3.4) is Ft -measurable and
continuous. Applying Schwarz inequality and Theorem 4.3.6 in [3] to (3.4) with n = 1, we have
t
t
1
0
2
0 2
E sup |x (s) − x (s)| 2(t − t0 )E
|B(s, xs )| ds + 8E
|σ(s, x0s )|2 ds, t ∈ [t0 , b].
t0 st
t0
t0
From (1.3), (3.1), (3.5) and (3.6), we obtain
E sup |x1 (s) − x0 (s)|2 C0 ,
t0 st
t ∈ [t0 , b].
For any t ∈ [t0 , b],
1
2
2
E sup |x (s)| 3E|ξ(0)| + 3E sup t st
t st
0
0
s
t0
2 B(s, x0s )ds
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0
s
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2 σ(s, x0s )dω(s) .
Notice that
sup
t0 −τ st
|x1 (s)|2 ξ2 + sup |x1 (s)|2 .
t0 st
Thus for any t ∈ [t0 , b]
E
s
s
2 2 |x1 (s)|2 4Eξ2 + 3E sup B(s, x0s )ds + 3E sup σ(s, x0s )dω(s)
t0 −τ st
t0 st
t0 st
t0
t0
t
t
|B(s, x0s )|2 ds + 12E
|σ(s, x0s )|2 ds
4Eξ2 + 3(b − t0 )E
sup
4Eξ2 + 3(b − t0 )E
t0
t
t0
t
+ 12E
t0
t0
(2|B(s, 0)|2 + 2Lx0s 2 )ds
(2|σ(s, 0)|2 + 2Lx0s 2 )ds
C0 (1 + Eξ2 ) + C0 E
t
t0
x0s 2 ds.
(3.12)
It leads to that
E
sup
t0 −τ st
|x1 (s)|2 C0 (1 + Eξ2 ) + C0 (t − t0 )Eξ2
C0 (1 + Eξ2 )eC0 (t−t0 ) ,
t ∈ [t0 , b].
So (3.7)–(3.9) hold for n = 1. Now, suppose (3.7)–(3.9) hold for n = k. Then for n = k + 1, from (1.3),
(3.1), (3.5), (3.6) and (3.9) with n = k, one obtains that
B(s, xks ) ∈ M2 ([t0 , b]; Rn ) and σ(s, xks ) ∈ M2 ([t0 , b]; Rn×m ).
(3.13)
Thus, from Theorem 4.3.2 and its remark in [3], xk+1 (t) defined by xk (t) and (3.4) is Ft -measurable and
continuous. We compute
t
E sup |xk+1 (s) − xk (s)|2 M̄ E
xks − xk−1
2 ds
s
t0 st
M̄
t0
t
t0
t
M̄
t0
E sup |xk (ζ) − xk−1 (ζ)|2 ds
t0 ζs
C0 [M̄ (s − t0 )]k−1
C0 [M̄ (t − t0 )]k
ds =
.
(k − 1)!
k!
(3.14)
Furthermore, imitating the proof of (3.12), we can get
E
sup
|x
k+1
t0 −τ st
(s)|
2
C0 (1 + Eξ ) + C0 E
2
t
t0
xks 2 ds,
t ∈ [t0 , b].
So, from (3.15) and (3.9) with n = k, one obtains that for any t ∈ [t0 , b],
E
sup
t0 −τ st
|xk+1 (s)|2 C0 (1 + Eξ2 ) + C0 E
sup |xks (θ)|2 ds
t0 −τ θ0
t
C0 (1 + Eξ2 ) + C0 E
sup
C0 (1 + Eξ2 ) + C0
t
t0 t0 −τ rs
t
|xk (r)|2 ds
C0 (1 + Eξ2 )eC0 (s−t0 ) ds
t0
(3.15)
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= C0 (1 + Eξ2 )eC0 (t−t0 ) .
(3.16)
That is, (3.7)–(3.9) hold for n = k + 1. Hence, by induction, (3.7)–(3.9) hold for all n 1. From (3.7),
we can then show in the same way as in the proof of Theorem 5.1.1 in [3, p. 98] and Theorem 2.3.1 in [10]
that xn (t) converges to x(t) in the sense of L2 as well as probability 1, and x(t) is a unique solution of
the initial problem (1.2) and (1.3).
On the other hand, (3.9) implies that xn (t) belongs to M2 ([t0 − τ, b]; Rn ). Letting n → ∞ and using
the Fatou’s Lemma [5], we obtain
(3.17)
E
sup |x(s)|2 C0 (1 + Eξ2 )eC0 (t−t0 ) , t ∈ [t0 , b].
t0 −τ st
This implies that x(t) belongs to M2 ([t0 − τ, b]; Rn ).
1
Remark 3.1. If B(t, x) = a(t) + tx, where a(t) = √
4
|t−1|
for t = 1 and a(1) = 0, then it is easy to
check that B(t, x) satisfies the first condition in (3.1). But it does not satisfy the condition (A), i.e., (1.7)
in [14], even the interval [t0 , b] including t = 1 is finite since B(t, 0) is unbounded in [t0 , b].
In Lemma 3.1, if the global Lipschitz condition on coefficients is relaxed to hold locally, we wonder if
there is a local solution? If so, what is the maximum existing interval of the local solution? The following
theorem will give the answers.
Theorem 3.1. Assume the condition (3.1) holds. If B and σ satisfy the local Lipschitz condition in
xt in [t0 , T ), then there must be a β ∈ (t0 , T ] such that the initial problem (1.2) and (1.3) has a unique
continuous solution x(t) for t ∈ [t0 − τ, β) and the solution x(t) explodes at β if β < T . Otherwise,
the solution x(t) exists globally in [t0 − τ, T ). In general, the solution x(t) given above is called to be a
non-continuable solution (see [19]).
Proof.
Since B and σ satisfy the local Lipschitz condition in xt in [t0 , T ), for any b ∈ (t0 , T ) and n > 0,
there exists a positive constant Kn such that
|B(t, φ) − B(t, ψ)| Kn φ − ψ,
|σ(t, φ) − σ(t, ψ)| Kn φ − ψ,
for all t ∈ [t0 , b] and φ, ψ ∈ C with φ ∨ ψ n. For the above n, define functions Bn and σn as follows:
n ∧ xt n ∧ xt xt , σn (t, xt ) = σ t,
xt ,
(3.18)
Bn (t, xt ) = B t,
xt xt t
where we set x
xt = 1 when xt ≡ 0. Then it is obvious that Bn and σn satisfy the global Lipschitz
condition (3.2) in [t0 , b] × C. By Lemma 3.1, the following equation
t
xn (t) = xn (t0 ) +
t
Bn (s, (xn )s )ds +
t0
(xn )t0 = xn (t0 + s) =
σn (s, (xn )s )dω(s),
t0 t b,
(3.19)
t0
ξ,
if ξ n,
0,
if ξ > n,
s ∈ [−τ, 0]
(3.20)
has a unique continuous solution xn (t).
Define a sequence of stopping time δn by
δn = b ∧ inf{t ∈ (t0 , b] : |xn (t)| n},
where we set inf ∅ = ∞ as usual. From (3.18) and (3.20), for t ∈ [t0 , δn ], we have known that
Bn+1 (s, (xn )s ) = Bn (s, (xn )s ) = B(s, (xn )s ),
σn+1 (s, (xn )s ) = σn (s, (xn )s ) = σ(s, (xn )s ).
(3.21)
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That is, (3.19) and the following equation
t
xn+1 (t) = xn+1 (t0 ) +
t
Bn+1 (s, (xn+1 )s )ds +
σn+1 (s, (xn+1 )s )dω(s),
t0
t0 t b,
(3.22)
t0
have the same coefficients for t ∈ [t0 , δn ] and their initial data overlap in D = {xt ∈ C : xt n}. Thus,
by the same proof of Theorem 5.2.1 in [3], we can get that
xn+1 (t) = xn (t),
t ∈ [t0 − τ, δn ], a.s.
This further implies that δn is increasing in n. So we can define δ = limn→∞ δn .
Now we suppose that
Ω(t) = {ω ∈ Ω : δ ∈ [t0 , t] ⊆ [t0 , b]},
(3.23)
β = sup {s ∈ [t0 , t] : P (Ω(s)) = 0}.
(3.24)
t0 tb
From the definition of β, there must be a sequence {t̄k : t̄k ∈ [β, b]} with limk→∞ t̄k = β such that
P (Ω(t̄k )) > 0.
(3.25)
For the above given t̄k , we can choose an integer Nt̄k satisfying
Δ
Nt̄k > N = Eξ2 ,
and P (Ω(t̄k ))Nt̄2k > N + 1.
Let I(·) denote the indicator function of (·) and
sup
Ωn (t) =
t0 −τ st
(3.26)
|xn (s)| > n .
Then, limn→∞ Ωn (t) = Ω(t) and P (Ω(t̄k )) P (ΩNt̄k (t̄k )). So we can get
E(IΩNt̄
k
(t̄k ) |xNt̄k (t̄k
∧ δNt̄k )|2 ) P (ΩNt̄k (t̄k ))Nt̄2k P (Ω(t̄k ))Nt̄2k .
(3.27)
Therefore, by using (3.26) and (3.27), we can obtain
E(|xNt̄k (t̄k ∧ δNt̄k )|2 ) E(IΩN
t̄k
(t̄k ) |xNt̄k (t̄k
∧ δNt̄k )|2 ) P (Ω(t̄k ))Nt̄2k > N + 1.
(3.28)
If β = t0 , noting that t0 t̄k ∧ δNt̄k t̄k , we have
t̄k ∧ δNt̄k → t0 ,
when t̄k → t0 .
This together with (3.28) implies that
E|ξ(0)|2 > N + 1 = Eξ2 + 1,
which is a contradiction. Therefore, we get β > t0 . Then, for any t̄ ∈ [t0 , β),
lim P (Ωn (s)) = 0 for s ∈ [t0 , t̄].
(3.29)
n→∞
For this case, we can prove x(t) defined by
x(t, ω) = lim xn (t, ω),
n→∞
∀ t ∈ [t0 , t̄] and ω ∈ lim Ωn (t̄)
n→∞
(3.30)
is the solution of (1.2) and (1.3). In fact, x(t, ω) = xn (t, ω) if ω ∈ Ωn (t̄) and t ∈ [t0 , t̄], and we get
xt n a.s. for ω ∈ Ωn (t̄), t ∈ [t0 , t̄].
(3.31)
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Combining with (3.21), we have for ω ∈ Ωn (t̄),
Bn (t, (xn )t ) = B(t, xt ),
σn (t, (xn )t ) = σ(t, xt ),
By using Lemma 4.2.11 in [3], for ω ∈ Ωn (t̄),
t
t
σ(s, xs )dω(s) =
σn (s, (xn )s )dω(s),
t0
a.s. ∀ t ∈ [t0 , t̄].
t0 t t̄, a.s.
(3.32)
(3.33)
t0
From (3.19), (3.32) and (3.33), we have for ω ∈ Ωn (t̄),
t
t
xn (t) = xn (t0 ) +
B(s, (xn )s )ds +
σ(s, (xn )s )dω(s),
t0
t0 t t̄ a.s.
(3.34)
t0
This implies x(t) = xn (t) for ω ∈ Ωn (t̄) is also the solution of (3.3) for t ∈ [t0 , t̄]. So, it is the solution
of (1.2) and (1.3) for ω ∈ Ωn (t̄) and t ∈ [t0 , t̄]. Combining (3.29), we get x(t) defined by (3.30) is the
solution to (1.2) and (1.3) almost for all ω ∈ Ω and t ∈ [t0 , t̄]. From the arbitrariness of taking t̄, the
solution x(t) exists in [t0 , β).
From the procedure of the above proof, it is obvious that the solution x(t) explodes at β defined by
(3.24) if there is a b > 0 such that β < b or β = b and P (Ω(b)) > 0. Otherwise, from the arbitrariness of
choosing b, the solution x(t) exists in [t0 , T ). Then, the proof is completed.
Remark 3.2.
Under the local Lipschitz condition in C space, Theorem 3.1 gives a similar result as
the Continuation Theorem in [19, Theorem 2]. It shows that under the conditions of Theorem 3.1, there
is always a non-continuable solution of (1.2) and (1.3), and the solution either explodes at a t ∈ (t0 , T )
or exists globally on [t0 − τ, T ).
Remark 3.3.
equation:
In Theorem 3.1, the condition (3.1) is necessary, for example, consider the following
dx(t) = [x(t) + α(t)]dt,
x(t0 ) = x0 , α(t) =
1
for t > t0 and α(t0 ) = 0.
t − t0
(3.35)
It is obvious that in (3.35) B(t, x) = x(t) + α(t) satisfies the local Lipschitz condition, but the condition
(3.1) is not satisfied. However, (3.35) does not have a non-continuable solution. Of course, Theorem 2.8
in [10, p. 154] can also not guarantee it to have a local solution.
Based on Theorem 3.1 and the differential comparison principle, the sufficient conditions for global
existence in [19] can also be obtained except that the local Lipschitz condition in L2 (Ω; C) is replaced by
the one in the sense of Definition 2.3 and the condition (3.1) and using the continuity of Ex(t). In the
following, we shall give some sufficient conditions on global existence without the continuity of Ex(t).
Corollary 3.1.
Let the conditions of Theorem 3.1 hold, τn the random variable equal the time at
which the process x(t) first leaves Un = {|x| < n}, and let τn (t) = τn ∧ t. If there is a function
V (t, x) ∈ C([t0 − τ, T ) × Rn ; R+ ) with lim|x|→∞ inf t0 s<T V (s, x) = ∞ such that for any n > 0 and any
t ∈ [t0 , T )
EV (τn (t), x(τn (t))) L(t),
(3.36)
where L : [t0 , T ) → R+ with sups∈[t0 ,t] L(s) < ∞ for any given t ∈ [t0 , T ), then β = T , that is, the
solution x(t) of (1.2) and (1.3) is unique and exists globally on [t0 − τ, T ).
Proof.
Using (3.36) and the Chebyshev’s inequality, for any t ∈ [t0 , T )
P (τn t) = P (|x(τn (t))| n) sups∈[t0 ,t] L(s)
inf s∈[t0 ,T ),|x|n V (s, x)
.
(3.37)
We claim that β = T . Otherwise, β < T and the solution x(t) explodes at β by Theorem 3.1. Noting
limn→∞ τn = δ defined in (3.23) and using sups∈[t0 ,t] L(s) < ∞ for any given t ∈ [β, T ) and (3.37), we
can get P (δ t) = 0, which is a contradiction. So x(t) exists globally on [t0 − τ, T ).
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Remark 3.4. It is obvious that (3.36) holds if the following priori estimate is satisfied
EV (t, x(t)) L(t).
(3.38)
If one chooses V (t, x) = xT x in Corollary 3.1, the global existence of the solution x(t) of (1.2) and (1.3)
can be implied by its mean square boundedness.
In the following, we can obtain a generalization of Hasminskii theorem in [7, p. 82].
Theorem 3.2. Let the conditions of Theorem 3.1 hold. Suppose that there are functions a(t) ∈
N 1 ([t0 , t]; R+ ) and b ∈ C([t0 , t]; R+ ) for any t ∈ [t0 , T ), and V ∈ C1,2 ([t0 , T ) × Rn ; R+ ) such that
inf V (t, x) = ∞,
(3.39)
lim
t0 t<T
|x|→∞
LV (t, x(t)) a(t) + b(t)V (t, x(t)),
∀ t ∈ [t0 , T ),
(3.40)
whenever V (t + s, x(t + s)) V (t, x(t)) for any s ∈ [−τ, 0]. Then there exists a unique global solution
x(t) to (1.2) and (1.3). Moreover, there exists a function L0 : [t0 , T ) → R+ with sups∈[t0 ,t] L0 (s) < ∞ for
any t ∈ [t0 , T ) such that
EV (t, x(t)) e
t
t0
b(s)ds
L0 (t).
(3.41)
That is, x(t) belongs to M2 ([t0 − τ, b]; Rn ) for any b ∈ [t0 , T ).
Proof.
From Corollary 3.1, we need to prove that the function V (τn (t), x(τn (t)) satisfies (3.36). Let
w(t, x(t)) = V (t, x(t))e
−
t
t0
b(s)ds
.
(3.42)
Then from Condition (3.40) and the continuity of b(t),
Lw(t, x) e
−
t
t0
b(s)ds
[a(t) + b(t)V − b(t)V )] = a(t)e
−
t
t0
b(s)ds
,
(3.43)
whenever V (t + s, x(t + s)) V (t, x(t)) for any s ∈ [−τ, 0].
Let w̄(t, x(t)) = sup−τ s0 w(t + s, x(t + s)) for t ∈ [t0 , T ), then for any ω ∈ Ω there is an s0 = s0 (ω) ∈
[−τ, 0] such that w̄(t, x(t)) = w(t + s0 , x(t + s0 )) and either s0 = 0 or s0 < 0 and w(t + s, x(t + s)) <
w(t + s0 , x(t + s0 )) for s0 < s 0. Let
D = {ω ∈ Ω : s0 (ω) = 0} and D̄ = Ω − D.
(3.44)
If ω ∈ D̄, then for h > 0 sufficiently small w̄(t + h, x(t + h)) = w̄(t, x(t)) and Lw̄(t, x(t)) = 0. If ω ∈ D,
then w̄(t, x(t)) = w(t, x(t)), that is, w(t, x) w(t + s, x(t + s)) for any s ∈ [−τ, 0], which implies that
V (t + s, x(t + s)) V (t, x(t)) for any s ∈ [−τ, 0]. From (3.43), we get Lw̄(t, x(t)) = Lw(t, x(t)) a(t)e
−
t
t0
b(s)ds
for ω ∈ D. Therefore, for almost all ω ∈ Ω, we always have
Lw̄(t, x(t)) a(t)e
−
t
−
t
t0
b(s)ds
,
b(s)ds
(3.45)
where a(t)e t0
∈ N 1 ([t0 , t]; R+ ) for any t ∈ [t0 , T ) can be implied by a ∈ N 1 ([t0 , t]; R+ ) and
b ∈ C([t0 , t]; R+ ) for any t ∈ [t0 , T ).
It is well known that the process x(τn (t)), obtained by stopping the process x(t) at the instant it
reaches the boundary of the domain Un , has an Itô differential. Therefore, by Itô’s formula, we can get
from (3.45) that
τn (t)
Ew̄(τn (t), x(τn (t))) = Ew̄(t0 , x(t0 )) + E
Lw̄(s, x(s))ds
t0
t0
sup V (t0 + s, x(t0 + s))
e t0 −τ b(s)ds E
−τ s0
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Δ
du = L0 (t).
(3.46)
t0
Thus, by (3.42), we can get
Since a(t)e
−
t
t0
EV (τn (t), x(τn (t)) e
b(s)ds
t
t0
b(s)ds
L0 (t).
(3.47)
∈ N 1 ([t0 , t]; R+ ) for any t ∈ [t0 , T ), we have
e
t
t0
b(s)ds
L0 (t) < ∞ for any given t ∈ [t0 , T ).
Consequently, by Corollary 3.1, the solution x(t) of (1.2) and (1.3) is unique and exists globally on
[t0 − τ, T ). The inequality (3.41) can be proved by letting n → ∞ in (3.47) and using Fatou’s lemma.
Thus, the proof is completed.
Remark 3.5. Theorem 3.2 is a natural generalization of Theorem 3.4.1 in [7] when τ ≡ 0, a(t) ≡ 0,
b is a constant, B and σ do not need the local linear growth condition. It is still valid if the inequality
(3.40) holds for all xt ∈ C with xt r for some positive constant r since one can easily see from the
proof of Theorem 3.1 that in this case the sample function cannot escape to infinity before it exits from
the set xt r.
4
Examples
To illustrate the efficiency of the results we obtained above, in this section, we will give two examples to
which the existing results cannot be applied.
Example 4.1. Consider the following stochastic differential equation with variable delays:
dx(t) = [α(t) + β(t)x(t − τ (t)) − x3 (t)]dt + γ(t)x(t − τ (t))dω(t),
t > t0 ,
(4.1)
where ω(t) is a scalar Brownian motion, α ∈ N 2 ([t0 , b]; R), β and γ ∈ C([t0 , b]; R) for any b ∈ (t0 , ∞) and
0 τ (t) τ (τ 0 is a constant).
It is easy to check that B(t, xt ) = α(t) + β(t)x(t − τ (t)) − x3 (t) and σ(t, xt ) = γ(t)x(t − τ (t)) satisfy all
the conditions of Theorem 3.1. Therefore, Theorem 3.1 yields that (4.1) with initial condition xt0 (s) =
ξ(s), s ∈ [−τ, 0] has a unique continuous non-continuable solution x(t). Furthermore, taking V (t, x) = x2 ,
we have
LV (t, x) 2x[α(t) + β(t)x(t − τ (t)) − x3 (t)] + γ 2 (t)x2 (t − τ (t))
α2 (t) + 2x2 + (β 2 (t) + γ 2 (t))x2 (t − τ (t))
a(t) + b(t)V (t, x(t)),
whenever x2 (t + s)) x2 (t),
∀ s ∈ [−τ, 0],
where a(t) = α2 (t) ∈ N 1 ([t0 , t]; R+ ) and b(t) = 2 + β 2 (t) + γ 2 (t) ∈ C([t0 , t]; R+ ) for any t ∈ (t0 , ∞).
Then, from Theorem 3.2, the solution of (4.1) and (1.3) uniquely exists on [t0 − τ, ∞).
1
Remark 4.1. If we take, for instance, α(t) = et or α(t) = √
4
|t−1|
for t = 1 and α(1) = 0, then B in
this example does not satisfy the condition (A). Thus the results in [14] are invalid for equation (4.1).
Example 4.2. We consider a non-autonomous stochastic Lotka-Volterra competitive system [8]
n
dxi (t) = xi (t) bi (t) −
aij (t)xj (t) dt + σi (t)dωi (t) ,
t > 0, i = 1, . . . , n,
(4.2)
j=1
with initial value xi (0) > 0, where ωi (t), 1 i n, are independent Brownian motions, bi , aij : R+ → R+
are Borel measurable functions with
max {bi (t), aij (t)} h(t),
1i,jn
∀ i, j = 1, . . . , n,
(4.3)
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i = 1, . . . , n.
(4.4)
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where h ∈ C(R+ ; R+ ) and
σi (t) ∈ N 4 ([0, b]; R) for any b ∈ (0, ∞),
First consider the following equation
n
1 2
yj (t)
dt + σi (t)dωi (t),
dyi (t) = bi (t) − σi (t) −
aij (t)e
2
j=1
t > 0,
(4.5)
with initial value yi (0) = ln xi (0), i = 1, . . . , n. Obviously, the coefficients of (4.5) satisfy the local
Lipschitz condition in y = (y1 , . . . , yn )T in [0, ∞). From (4.3), for (4.5) we have
2
n
1 2
2
|Bi (t, 0)| = bi (t) − σi (t) −
aij (t) [(n + 1)h(t) + σi2 (t)]2 .
2
j=1
Then, Condition (3.1) is easily implied by (4.4) and the continuity of h(t). Thus, there is a unique noncontinuable solution y(t) for t ∈ [0, β), where β is the right endpoint of the maximum existing interval
of y(t). Therefore, by Itô’s formula, it is easy to see xi (t) = eyi (t) is the unique positive solution to (4.2)
with initial value xi (0) > 0, i = 1, . . . , n.
Next, we will show this non-continuable solution is global, i.e., β = ∞. To show this statement, we
define V : Rn → R+ by
n
V (y) =
[eyi − 1 − yi ], y = (y1 , . . . , yn )T .
i=1
It is obvious that eyi − 1 − yi 0 for yi ∈ R and V (y) → ∞ as |y| → ∞.
From (4.3), we have
n n
1
1
LV (y) =
(eyi − 1) bi (t) − σi2 (t) −
aij (t)eyj + σi2 (t)eyi
2
2
i=1
j=1
n
n
1
bi (t)eyi +
aij (t)eyj + σi2 (t)
2
i=1
j=1
(1 + n)h(t)
n
i=1
1 2
σ (t).
2 i=1 i
n
ey i +
Noting that e 2[e − 1 − u] + 2 ln 2 for u ∈ R, we know that
u
u
1 2
σ (t)
2 i=1 i
n
LV (y) 2(n + 1)h(t)[V (y) + n ln 2] +
1 2
σ (t)
2 i=1 i
n
2(n + 1)h(t)V (y) + 2n(n + 1) ln 2h(t) +
a(t) + b(t)V (y),
n
where a(t) = 2n(n + 1) ln 2h(t) + 12 i=1 σi2 (t) ∈ N 1 ([0, b]; R+ ) for any b ∈ (0, ∞), b(t) = 2(n + 1)h(t) ∈
C([0, b]; R+ ). Then, from Theorem 3.2, the solution y ∈ Rn of (4.5) uniquely exists on R+ . So the
solution x(t) ∈ Rn+ − {0} of (4.2) uniquely exists on R+ .
Remark 4.2. Example 4.2 is a better result than Theorem 2.1 in [8] which requires the boundedness
and continuity of the coefficients of (4.2). Of course, Theorem 3.4.1 in [7] is also invalid for Example 4.2
by the same token.
Acknowledgements This work was supported by National Natural Science Foundation of China (Grant Nos.
11271270, 11201320 and 11101298) and Youth Foundation of Sichuan University (Grant No. 2011SCU11111). The
authors would like to thank the reviewers for their constructive suggestions and valuable comments. Moreover,
we should thank Dr. Lingying Teng for her helpful comments and suggestions.
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