The uncertain measure
is a real-valued set function on a σ-algebra
over a nonempty set Γ satisfying normality, duality, subadditivity, and product axioms. The triplet
is called an uncertainty space.
Definition 1
(
[8]) An uncertain variable is a measurable function from an uncertainty space
to the set of real numbers, i.e., for any Borel set B of real numbers, the set
is an event.
The uncertainty distribution function Φ : ℜ → [0,1] of an uncertain variable ξ is defined as
. The expected value of an uncertain variable is defined as follows.
Definition 2
(
[8]) Let ξ be an uncertain variable. Then the expected value of ξ is defined by
provided that at least one of the two integrals is finite.
The expected value can also be written as
where Φ(r) is the uncertainty distribution of ξ. If ξ is an uncertain variable with finite expected value e, then the variance of ξ is defined as Var[ξ] = E[(ξ−e)2].
Let ξ
1,ξ
2,⋯,ξ
n
be independent uncertain variables with uncertainty distributions Φ1,Φ2,⋯,Φ
n
, respectively. Liu
[26] showed that if the function f(x
1,x
2,⋯,x
n
) is strictly increasing with respect to x
1,x
2,⋯, x
m
and strictly decreasing with respect to x
m+1,x
m+2,⋯,x
n
, then
is an uncertain variable with inverse uncertainty distribution
(1)
Furthermore, Liu and Ha
[27] proved that it has an expected value
(2)
Definition 3
(
[9]) Let T be an index set and let
be an uncertainty space. An uncertain process is a measurable function from
to the set of real numbers, i.e., for each t ∈ T and any Borel set B of real numbers, the set
is an event.
Definition 4
(
[9]) Let ξ
1,ξ
2,⋯ be iid positive uncertain variables. Define
for n ≥ 1. Then the uncertain process
is called a renewal process.
An uncertain process X
t
is said to have independent increments if
are independent uncertain variables where t
0 is the initial time and t
1,t
2,⋯,t
k
are any times with t
0 < t
1 < ⋯ < t
k
. An uncertain process X
t
is said to have stationary increments if, for any given t > 0, the increments X
s+t
− X
s
are identically distributed uncertain variables for all s > 0. An uncertain process S
t
is said to be a stationary independent increment process if it has stationary and independent increments. Liu
[26] proved that the expected value of stationary independent increment process S
t
is E[S
t
] = a + bt.
Definition 5
(
[12]) An uncertain process C
t
is said to be a canonical Liu process if:
1. C
0 = 0 and almost all sample paths are Lipschitz continuous.
2. C
t
has stationary and independent increments.
3. Every increment C
s+t
− C
s
is a normal uncertain variable with expected value 0 and variance t2, whose uncertainty distribution is
Liu integral
In order to deal with the integration and differentiation of uncertain processes, Liu
[12] proposed an uncertain integral with respect to the Liu process.
Definition 6
(
[12]) Let X
t
be an uncertain process and C
t
be a canonical Liu process. For any partition of closed interval [a,b] with a = t
1 < t
2 < ⋯ < t
k+1 = b, the mesh is written as
Then the Liu integral of X
t
is defined as
provided that the limit exists almost surely and is finite. In this case, the uncertain process X
t
is said to be Liu integrable.
Liu
[12] verified the fundamental theorem of uncertain calculus, i.e., for a canonical Liu process C
t
and a continuously differentiable function h(t,c), the uncertain process Z
t
= h(t,C
t
) has a differential
Yao integral
In order to study an uncertain integral with respect to an uncertain process admitting jumps, an uncertain integral with respect to a renewal process was introduced by Yao
[14].
Definition 7
(
[14]) Let X
t
be an uncertain process and N
t
be an uncertain renewal process. For any partition of closed interval [a,b] with a = t
1 < t
2 < ⋯ < t
k+1 = b, the mesh is written as
Then the Yao integral of X
t
is defined as
provided that the limit exists almost surely and is finite. In this case, the uncertain process X
t
is said to be Yao integrable.
Yao
[14] verified the fundamental theorem of uncertain calculus, i.e., for a renewal process N
t
and a continuously differentiable function h(t,n), the uncertain process Z
t
= h(t,N
t
) has a differential
Uncertain integral with finite variation processes
Definition 8
(
[15]) Let X
t
be an uncertain process and let A
t
be a finite variation process. For any partition of closed interval [a,b] with a = t
1 < t
2 < ⋯ < t
k+1 = b, the mesh is written as
Then the uncertain integral of X
t
with respect to A
t
is
provided that the limit exists almost surely and is finite. In this case, the uncertain process X
t
is said to be integrable with respect to the finite variation process A
t
.
Suppose that A
t
is a finite variation process and h(t,s) a continuously differentiable function, the uncertain process Z
t
= h(t,A
t
) has a differential
Uncertain differential equation
Definition 9
(
[9]) Suppose C
t
is a canonical Liu process, and f and g are some given functions. Given an initial value X
0, then
(3)
is called an uncertain differential equation with an initial value X
0. A solution is an uncertain process X
t
that satisfies (3) identically in t.
Theorem 1
(Existence and Uniqueness Theorem [16]) The uncertain differential equation (3) has a unique solution if the coefficients f(x,t) and g(x,t) satisfy the Lipschitz condition
and linear growth condition
for some constant L. Moreover, the solution is sample-continuous.
Definition 10
(
[20]) The α-path (0 < α < 1) of an uncertain differential equation
with initial value X
0 is a deterministic function
with respect to t that solves the corresponding ordinary differential equation
where Φ−1(α) is the inverse uncertainty distribution of the standard normal uncertain variable, i.e.,
Theorem 2
(Yao-Chen Formula
[20]) Let X
t
and
be the solution and α-path of the uncertain differential equation
respectively. Then
The Yao-Chen formula relates an uncertain differential equation and a family of ordinary differential equations just like the Feyman-Kac formula relates a stochastic differential equation and a partial differential equation. Besides, Yao
[28] studied the integral of solution to uncertain differential equations.