Soft Comput
DOI 10.1007/s00500-014-1521-4
METHODOLOGIES AND APPLICATION
Almost sure stability for uncertain differential equation
with jumps
Xiaoyu Ji · Hua Ke
© Springer-Verlag Berlin Heidelberg 2014
Abstract Uncertain differential equation with jumps, as a
crucial tool to deal with a discontinuous uncertain system,
is a type of differential equation driven by both canonical
Liu process and uncertain renewal process. So far, a concept
of stability in measure for an uncertain differential equation with jumps has been proposed. As a supplement, this
paper proposes a concept of almost sure stability for an uncertain differential equation with jumps. A sufficient condition
is derived for an uncertain differential equation with jumps
being stable almost surely. As a corollary, a sufficient condition is also given for a linear uncertain differential equation
with jumps being stable almost surely.
Keywords Uncertain differential equation · Stability ·
Almost sure stability · Uncertainty theory
1 Introduction
Stochastic differential equation driven by Wiener process
was first proposed by Ito in 1940s to deal with dynamic stochastic systems. Since then, it has been widely studied and
applied in many areas. For example, Kalman and Bucy (1961)
proposed a method to filter the noises from observed data
via stochastic differential equation, and Black and Scholes
Communicated by V. Loia.
X. Ji
School of Business, Renmin University of China,
Beijing 100872, China
e-mail: [email protected]
H. Ke (B)
School of Economics and Management, Tongji University,
Shanghai 200092, China
e-mail: [email protected]
(1973) assumed the price of a stock followed a stochastic differential equation, and derived the famous European option
pricing formulas for the stock. As a supplement of Ito’s stochastic differential equation, stochastic differential equation
with jumps, which essentially is a type of differential equation driven by both Wiener process and Poisson process, also
draws attention from the researchers. It is mainly used to
describe a stochastic differentiable system with sudden drifts.
Except for randomness, human uncertainty is another
source of indeterminate information. To deal with human
uncertainty, an uncertainty theory was founded by Liu (2007)
and refined by Liu (2010) based on normality axiom, duality
axiom, subadditivity axiom, and product axiom. A concept of
uncertain variable was first defined by Liu (2007) to model
a quantity under uncertainty, and a concept of uncertainty
distribution was also proposed to describe an uncertain variable in practice. Peng and Iwamura (2010) gave a sufficient
condition for a function being an uncertainty distribution of
an uncertain variable. Yao and Ji (2014) proposed a decisionmaking method by means of uncertain variables in an uncertain environment.
To describe an uncertain system evolving with time, Liu
(2008) proposed a concept of uncertain process as a sequence
of uncertain variables driven by the time. After that, Liu
(2009) designed a canonical Liu process as a counterpart
of standard Wiener process. Canonical Liu process is a type
of uncertain process with stationary and independent normal
increments, and almost all its sample paths are Lipschitz continuous. Besides, Liu (2009) founded an uncertain calculus
with respect to a canonical Liu process, which was generalized by Liu and Yao (2012) to a case with multiple canonical
Liu processes.
Uncertain differential equation driven by canonical Liu
process is used to describe a continuous and dynamic uncertain system with some change rules. So far, it has been widely
123
X. Ji, H. Ke
applied to the financial markets. Liu (2009) assumed the stock
price followed a geometric Liu process, and proposed a stock
model with human uncertainty. Then, Chen (2011) derived
the American option pricing formulas for Liu’s stock model.
Besides, Chen and Gao (2013) studied some term structure
models of interest rate, and Jiao and Yao (2014) derived the
price of zero-coupon bond for a type of term structure model.
In addition, Liu et al. (2014b) proposed an uncertain currency
exchanging model. For the recent development of uncertain
finance, interested readers may refer to Liu (2013). In theoretical aspects, Chen and Liu (2010) gave a sufficient condition for an uncertain differential equation having a unique
solution, and Yao et al. (2013) gave a sufficient condition for
an uncertain differential equation being stable in measure.
In addition, Liu et al. (2014a) gave a sufficient condition of
almost sure stability.
To describe a discontinuous uncertain system, Liu (2008)
proposed an uncertain renewal process, which was further
studied by Yao and Li (2012) and Zhang et al. (2013). Yao
(2012) founded an uncertain calculus with respect to uncertain renewal process, and proposed an uncertain differential
equation with jumps, which is essentially a type of differential equation driven by both canonical Liu process and
uncertain renewal process. Based on the uncertain differential equation with jumps, Yu (2012) presented an uncertain
stock model with jumps. Recently, Yao (2014) solved two
types of uncertain differential equations with jumps, and gave
a sufficient condition for it having a unique solution.
In this paper, we will propose a concept of almost sure
stability for an uncertain differential equation with jumps,
and give its sufficient condition. The rest of this paper is
organized as follows. In Sects. 2 and 3, we will introduce
uncertain variable and uncertain differential equation with
jumps, respectively. In Sect. 4, we will define almost sure
stability for an uncertain differential equation with jumps,
and give two examples to explain the definition. Then in
Sect. 5, we will give a sufficient condition for an uncertain
differential equation with jumps being stable almost surely.
As a corollary, a sufficient condition for a linear uncertain
differential equation with jumps will also be derived. At last,
some remarks are made in Sect. 6.
Axiom 1: (Normality Axiom) M{} = 1 for the universal
set .
Axiom 2: (Duality Axiom) M{} + M{c } = 1 for any
event .
Axiom 3: (Subadditivity Axiom) For every countable
sequence of events 1 , 2 , . . . , we have
∞
∞
M
i ≤
M {i } .
i=1
i=1
Besides, the product uncertain measure on a product σ algebra L was defined by Liu (2009) as follows:
Axiom 4: (Product Axiom) Let (k , Lk , Mk ) be uncertainty spaces for k = 1, 2, . . .. Then, the product uncertain
measure M is an uncertain measure satisfying
∞
∞
k =
Mk {k }
M
k=1
k=1
where k are arbitrarily chosen events from Lk for k =
1, 2, . . ., respectively.
An uncertain variable ξ is a measurable function from
an uncertainty space (, L, M) to the real number set .
To describe an uncertain variable in practice, a concept
of uncertainty distribution was defined by Liu (2007) as
(x) = M{ξ ≤ x} for any real number x. The inverse
function −1 (α) is called an inverse uncertainty distribution
of ξ if it exists and is unique for each α ∈ (0, 1). The expected
value of an uncertain variable ξ is defined by
0
+∞
M{ξ ≥ r }dr −
M{ξ ≤ r }dr
E[ξ ] =
provided that at least one of the two integrals is finite. For an
uncertain variable ξ with a regular uncertainty distribution
, Liu (2007) proved that
+∞
0
1
E[ξ ] =
(1−(r ))dr −
(r )dr =
−1 (α)dα.
In this section, we introduce some concepts about uncertain
variable, including uncertainty distribution, expected value,
independence, and operational law.
Definition 1 (Liu 2007) Let be a universal set, and L be a
σ -algebra on . A set function M : L → [0, 1] is called an
uncertain measure if it satisfies the following axioms:
123
−∞
0
0
Definition 2 (Liu 2009) The uncertain variables ξ1 , ξ2 , . . . ,
ξm are said to be independent if
m
m
M
(ξi ∈ Bi ) =
M{ξi ∈ Bi }
i=1
2 Uncertain variable
−∞
0
k=1
for any Borel sets B1 , B2 , . . . , Bm of real numbers.
Theorem 1 (Operational Law, Liu 2010) Let ξ1 , ξ2 , . . . , ξn
be independent uncertain variables with regular uncertainty distributions 1 , 2 , . . . , n , respectively. If the function f (x1 , x2 , . . . , xn ) is strictly increasing with respect
to x1 , x2 , . . . , xm and strictly decreasing with respect to
xm+1 , xm+2 , . . . , xn , then
ξ = f (ξ1 , ξ2 , . . . , ξn )
Almost sure stability for uncertain differential equation
is an uncertain variable with an inverse uncertainty distribution
−1
−1 (α) = f (−1
1 (α),. . . ,m (α),
−1
−1
m+1 (1 − α), . . . , n (1 − α)).
interval [a, b] with a = t1 < t2 < · · · < tk+1 = b, the mesh
is written as:
= max |ti+1 − ti |.
1≤i≤k
Then, the Liu integral of X t is defined by
3 Uncertain differential equation with jumps
b
X t dCt = lim
→0
a
In this section, we first introduce the uncertain calculus theory with respect to canonical Liu process and with respect
to uncertain renewal process. Then, we introduce uncertain
differential equation with jumps and its existence and uniqueness theorem.
Definition 3 (Liu 2008) Let T be an index set, and let
(, L, M) be an uncertainty space. An uncertain process
is a measurable function from T × (, L, M) to the set of
real numbers, i.e., for each t ∈ T and any Borel set B of real
numbers, the set
{X t ∈ B} = {γ | X t (γ ) ∈ B}
is an event.
Definition 4 (Liu 2009) An uncertain process Ct is said to
be a canonical Liu process if
(i) C0 = 0 and almost all sample paths are Lipschitz continuous,
(ii) Ct has stationary and independent increments,
(iii) every increment Cs+t −Cs is a normal uncertain variable
with an uncertainty distribution
−1
πx
t (x) = 1 + exp − √
, x ∈ .
3t
Different from Wiener process, almost all the sample paths
of a canonical Liu process are Lipschitz continuous. Yao et
al. (2013) regarded Lipschitz constants of the sample paths
as different values that an uncertain variable may take, and
proved the following theorem.
Theorem 2 (Yao et al. 2013) Let Ct be a canonical Liu
process on an uncertainty space (, L, M). Then, there
exists an uncertain variable K such that K (γ ) is a Lipschitz
constant of the sample path Ct (γ ) for each γ , and
lim M{γ ∈ | K (γ ) ≤ x} = 1.
x→+∞
By means of canonical Liu process, a Liu integral is
defined as an uncertain counterpart of Ito integral as below.
Definition 5 (Liu 2009) Let X t be an uncertain process and
Ct be a canonical Liu process. For any partition of closed
k
X ti · (Cti+1 − Cti )
i=1
provided that the limit exists almost surely and is finite.
Definition 6 (Liu 2008) Let ξ1 , ξ2 , . . . be iid positive uncertain variables. Define S0 = 0 and Sn = ξ1 + ξ2 + · · · + ξn
for n ≥ 1. Then, the uncertain process
Nt = max{n|Sn ≤ t}
n≥0
is called an uncertain renewal process.
Definition 7 (Yao 2012) Let X t be an uncertain process and
Nt be an uncertain renewal process. Then, the Yao integral
of X t on the interval [a, b] is defined by
b
X t dNt =
X t− (Nt − Nt− )
a
a<t≤b
provided that the sum exists almost surely and is finite.
Yao (2011) derived the fundamental theorem of uncertain calculus with respect to both canonical Liu process and
uncertain renewal process.
Definition 8 (Yao 2011) Let Ct be a canonical Liu process,
Nt be an uncertain renewal process, and h(t, c, n) be a continuously differentiable function. Then, the uncertain process
Z t = h(t, Ct , Nt ) can be represented by
s
s
∂h
∂h
Zs = Z0 +
(t, Ct , Nt )dt +
(t, Ct , Nt )dCt
∂t
0 ∂c
s0
+
(h(t, Ct , Nt ) − h(t, Ct , Nt− ))dNt ,
0
i.e., it has an uncertain differential
dZ t =
∂h
∂h
(t, Ct , Nt )dt +
(t, Ct , Nt )dCt
∂t
∂c
+(h(t, Ct , Nt ) − h(t, Ct , Nt− ))dNt .
Definition 9 (Yao 2012) Suppose that Ct is a canonical Liu
process, Nt is an uncertain renewal process, and f, g and h
are some given real functions. Then
dX t = f (t, X t )dt + g(t, X t )dCt + h(t, X t )dNt
is called an uncertain differential equation with jumps.
123
X. Ji, H. Ke
Example 1 Let Ct be a canonical Liu process, Nt be an uncertain renewal process with iid uncertain interarrival times
ξ1 , ξ2 , . . . , and Ut , Vt and Rt be some real functions. Then,
the uncertain differential equation with jumps
dX t = Ut X t dt + Vt X t dCt + Rt X t dNt
0
Yt (γ ) = Y0 + μt + σ Ct (γ ) + ν Nt (γ ), ∀γ ∈ ,
respectively. Since
|X t (γ ) − Yt (γ )| = |X 0 − Y0 |, ∀t ≥ 0, γ ∈ ,
has a solution
t
t
Nt
X t = X 0 exp
Us ds +
Vs dCs
(1 + R Si )
0
and
we have
sup |X t (γ ) − Yt (γ )| = |X 0 − Y0 |, ∀γ ∈ i=1
t≥0
where S0 = 0 and Si = ξ1 + ξ2 + · · · + ξi for i ≥ 1.
Theorem 3 (Yao 2014) The uncertain differential equation
with jumps
and
M γ ∈ sup |X t (γ ) − Yt (γ )| = 0
lim
|X 0 −Y0 |→0 t≥0
= M γ ∈ dX t = f (t, X t )dt + g(t, X t )dCt + h(t, X t )dNt
has a unique solution if the coefficients f (t, x) and g(t, x)
satisfy the linear growth condition
| f (t, x)| + |g(t, x)| ≤ L(1 + |x|), ∀x ∈ , t ≥ 0
lim
|X 0 −Y0 |→0
|X 0 − Y0 | = 0 = 1.
Hence the uncertain differential equation with jumps (2) is
stable almost surely.
Example 3 Consider a homogeneous uncertain differential
equation with jumps
and the Lipschitz condition
| f (t, x) − f (t, y)| + |g(t, x) − g(t, y)| ≤ L|x − y|,
∀x, y ∈ , t ≥ 0
dX t = μX t dt + σ X t dCt + ν X t dNt
for a constant L, and the coefficient h(t, x) is a real-valued
function.
(3)
where μ, σ, ν are positive real numbers. Note that its solutions with different initial values X 0 and Y0 are
X t (γ ) = X 0 · exp(μt + σ Ct (γ )) · (1 + ν) Nt (γ ) , ∀γ ∈ 4 Almost sure stability
and
In this section, we propose a concept of almost sure stability
for an uncertain differential equation with jumps, and give
two examples to explain the definition.
Definition 10 Let X t and Yt be two solutions of the uncertain
differential equation with jumps
dX t = f (t, X t )dt + g(t, X t )dCt + h(t, X t )dNt
(1)
with different initial values X 0 and Y0 , respectively. Then the
uncertain differential equation (1) is said to be stable almost
surely if
M γ ∈ lim sup |X t (γ ) − Yt (γ )| = 0 = 1.
|X 0 −Y0 |→0 t≥0
Example 2 Consider a linear uncertain differential equation
with jumps
dX t = μdt + σ dCt + νdNt .
(2)
Note that its solutions with different initial values X 0 and Y0
are
X t (γ ) = X 0 + μt + σ Ct (γ ) + ν Nt (γ ), ∀γ ∈ 123
Yt (γ ) = Y0 · exp(μt + σ Ct (γ )) · (1 + ν) Nt (γ ) , ∀γ ∈ ,
respectively. Since
|X t (γ ) − Yt (γ )| = |X 0 − Y0 | · exp(μt + σ Ct (γ ))
·(1 + ν) Nt (γ ) → ∞
as t → ∞ for each sample γ with Ct (γ ) > 0, we have
M γ ∈ lim sup |X t (γ ) − Yt (γ )| = 0 < 1.
|X 0 −Y0 |→0 t≥0
Hence the uncertain differential equation with jumps (3) is
not stable almost surely.
5 Sufficient condition
In this section, we give a sufficient condition for an uncertain
differential equation with jumps being stable almost surely.
As a corollary, we also give a sufficient condition for a linear uncertain differential equation with jumps being stable
almost surely.
Almost sure stability for uncertain differential equation
Theorem 4 The uncertain differential equation with jumps
dX t = f (t, X t )dt + g(t, X t )dCt + h(t, X t )dNt
+
t
L 2 (s)|X s (γ ) − Ys (γ )|dNs (γ ).
0
(4)
is stable almost surely if the coefficients f (t, x) and g(t, x)
satisfy
Let ξ1 , ξ2 , . . . denote the iid positive uncertain interarrival
times of Nt . Write S0 = 0 and Si = ξ1 + ξ2 + · · · + ξi for
i ≥ 1. Then, according to the Grownwall inequality, we have
| f (t, x) − f (t, y)| + |g(t, x) − g(t, y)| ≤ L 1 (t)|x − y|,
|X t (γ ) − Yt (γ )|
∀x, y ∈ , t ≥ 0
for some integrable function L 1 (t) on [0, +∞), and the coefficient h(t, x) satisfies
|h(t, x) − h(t, y)| ≤ L 2 (t)|x − y|, ∀x, y ∈ , t ≥ 0
for some monotone and integrable function L 2 (t) on [0, +∞).
≤ |X 0 −Y0 |·exp (1+ K (γ ))
t
N
t (γ )
L 1 (s)ds ·
(1+ L 2 (Si (γ )))
0
≤ |X 0 −Y0 |·exp (1+ K (γ ))
i=1
+∞
∞
L 1 (s)ds · (1+ L 2 (Si (γ )))
0
≤ |X 0 −Y0 |·exp (1+ K (γ ))
+∞
i=1
L 1 (s)ds ·exp
∞
0
Proof Let X t be a solution of the uncertain differential equation with jumps (4) with an initial value X 0 , and Yt be a
solution with an initial value Y0 , i.e.,
t
t
X t (γ ) = X 0 +
f (s, X s (γ ))ds +
g(s, X s (γ ))dCs (γ )
0
0
t
+ h(s, X s (γ ))dNs (γ ), ∀γ ∈ ,
0
t
t
Yt (γ ) = Y0 +
f (s, Ys (γ ))ds +
g(s, Ys (γ ))dCs (γ )
0
0
t
+ h(s, Ys (γ ))dNs (γ ), ∀γ ∈ .
sup |X t (γ ) − Yt (γ )| ≤ |X 0 − Y0 |
t≥0
+∞
· exp (1+ K (γ ))
We first consider
exp (1 + K (γ ))
+∞
L 2 (Si (γ )) .
i=1
L 1 (s)ds .
0
|X t (γ ) − Yt (γ )|
Since
+∞
0
L 1 (s)ds · exp
∞
(5)
By Theorem 2, we have
t
0
Assume K (γ ) is a Lipschitz constant of the sample path
Ct (γ ). Then, we have
| f (s, X s (γ )) − f (s, Ys (γ ))|ds
≤ |X 0 − Y0 | +
0
t
+
|g(s, X s (γ ))−g(s, Ys (γ ))|·|dCs (γ )|
0
t
+
|h(s, X s (γ ))−h(s, Ys (γ ))|dNs (γ )
0
t
≤ |X 0 − Y0 | +
L 1 (s)|X s (γ ) − Ys (γ )|ds
0
t
+
L 1 (s)|X s (γ ) − Ys (γ )| · |dCs (γ )|
0
t
+
L 2 (s)|X s (γ ) − Ys (γ )|dNs (γ )
0
t
≤ |X 0 − Y0 | +
L 1 (s)|X s (γ ) − Ys (γ )|ds
0
t
+ K (γ )
L 1 (s)|X s (γ ) − Ys (γ )|ds
0
t
+
L 2 (s)|X s (γ ) − Ys (γ )|dNs (γ )
0
t
= |X 0 − Y0 | + (1 + K (γ ))
L 1 (s)|X s (γ ) − Ys (γ )|ds
L 2 (Si (γ ))
i=1
for all t ≥ 0. Thus
0
M{γ ∈ | K (γ ) < +∞} = 1.
L 1 (s)ds < +∞,
0
we have
M γ ∈ exp (1+ K (γ ))
+∞
L 1 (s)ds
< +∞ = 1.
0
(6)
Now, we consider
∞
exp
L 2 (Si (γ )) .
i=1
Since
∞
L 2 (Si (γ ))· inf ξ j (γ ) ≤
j≥1
i=1
≤
∞
L 2 (Si (γ ))·ξi+1 (γ )
i=1
+∞
L 2 (s)ds,
0
we have
∞
i=1
L 2 (Si (γ )) ≤
1
inf ξ j (γ )
j≥1
+∞
L 2 (s)ds.
0
123
X. Ji, H. Ke
Noting that
M γ ∈ inf ξ j (γ ) > 0 = 1
we take
L 1 (t) = |μ1t | + |σ1t |
which is integrable on + . Since
j≥1
and
+∞
|h(t, x) − h(t, y)| = |ν1t ||x − y|,
L 2 (s) < +∞,
we take
0
L 2 (t) = |ν2t |
we have
∞
L 2 (Si (γ )) < +∞ = 1.
M γ ∈ exp
(7)
i=1
By Eqs. (5), (6), and (7), we have
M γ ∈ lim sup |X t (γ ) − Yt (γ )| = 0 = 1,
6 Conclusions
|X 0 −Y0 |→0 t≥0
and the uncertain differential equation with jumps (4) is stable
almost surely according to Definition 10. This completes the
proof.
Example 4 Consider an uncertain differential equation with
jumps
dX t = t 2 dt + exp(−t 2 − X t2 )dCt + exp(−t)X t dNt .
which is monotone and integrable on + . By Theorem 4,
the uncertain differential equation with jumps (9) is stable
almost surely.
(8)
Since f (t, x) = t 2 and g(t, x) = exp(−t 2 − x 2 ) satisfy
| f (t, x) − f (t, y)| + |g(t, x) − g(t, y)| ≤ exp(−t 2 )|x − y|
with respect to an integrable function exp(−t 2 ) on + for
all x, y ∈ , and h(t, x) = exp(−t)x satisfies
|h(t, x)−h(t, y)| = | exp(−t)x − exp(−t)y|
Uncertain differential equation with jumps is a type of differential equation driven by both canonical Liu process and
uncertain renewal process. This paper proposed a concept
of almost sure stability for an uncertain differential equation with jumps. A sufficient condition for an uncertain differential equation with jumps being stable almost surely
was derived. As a corollary, a sufficient condition for a linear uncertain differential equation with jumps being stable
almost surely was also obtained. In addition, these sufficient conditions were illustrated by some examples. Further
researches may cover the applications of uncertain differential equation with jumps in the area of finance and optimal
control.
Acknowledgments This work was supported by National Natural Science Foundation of China (Grant No. 71171191, Grant No.
71371141, and Grant No. 71001080).
= exp(−t)|x − y|
with respect to a monotone and integrable function exp(−t)
on + for all x, y ∈ , the uncertain differential equation
with jumps (8) is stable almost surely.
Corollary 1 The linear uncertain differential equation with
jumps
dX t = (μ1t X t + μ2t )dt + (σ1t X t + σ2t )dCt
+(ν1t X t + ν2t )dNt
(9)
is stable almost surely if |ν1t | is monotone, and
+∞
+∞
+∞
|μ1t |dt +
|σ1t |dt +
|ν1t |dt < +∞.
0
0
0
Proof Note that for the linear uncertain differential equation
with jumps (9), we have f (t, x) = μ1t x + μ2t , g(t, x) =
σ1t x + σ2t , and h(t, x) = ν1t x + ν2t . Since
| f (t, x) − f (t, y)| + |g(t, x) − g(t, y)|
= |μ1t ||x − y| + |σ1t ||x − y| = (|μ1t | + |σ1t |)|x − y|,
123
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