Uncertain calculus with renewal process

Fuzzy Optim Decis Making (2012) 11:285–297
DOI 10.1007/s10700-012-9132-y
Uncertain calculus with renewal process
Kai Yao
Published online: 27 April 2012
© Springer Science+Business Media, LLC 2012
Abstract Uncertain calculus is a branch of mathematics that deals with differentiation
and integration of function of uncertain processes. As a fundamental concept,
uncertain integral has been defined with respect to canonical process. However,
emergencies such as economic crisis and war occur occasionally, which may cause
the uncertain process a sudden change. So far, uncertain renewal process has been
employed to model these jumps. This paper will present a new uncertain integral
with respect to renewal process. Besides, this paper will propose a type of uncertain
differential equation driven by both canonical process and renewal process.
Keywords Uncertain calculus · Uncertain integral · Renewal process ·
Uncertain differential equation · Uncertainty theory
1 Introduction
Brownian motion is named after the Scottish biologist Brown who first observed the
motion of pollen in the water in 1827. After that, it was studied by many researchers,
and modeled by Wiener (1923) in 1923 as a continuous stochastic process with stationary
and independent increments which follow a normal distribution. In mid-twentieth
century, Ito (1944) founded stochastic calculus to deal with integration and differentiation
of a stochastic process with respect to Brownian motion. Following that, stochastic
differential equation, a type of differential equation driven by Brownian motion, was
studied and applied widely. In filtration, Kalman and Bucy (1961) applied stochastic
differential equation to filtering the noise away from the observations in 1961.
K. Yao (B)
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
e-mail: yaok09@mails.tsinghua.edu.cn
123

286 K. Yao
In finance, Black and Scholes (1973) assumed that the stock price follows a geometric
Brownian motion and proposed the famous Black–Scholes stock model in 1977.
Randomness sometimes exists with jumps in economics and natural sciences, for
example, emergency occurs occasionally in daily life, which may cause the stock price
a sudden shift. In order to describe the jumps, researchers proposed stochastic calculus
with respect to Poisson process as a supplement of Ito calculus. Besides, stochastic
differential equation driven by both Wiener process and Poisson process was also
proposed.
Although randomness has been used to describe undetermined properties for a long
time, some imprecise quantities, such as information and knowledge represented by
human language, do not behave like randomness. In order to model these imprecise
quantities, an uncertainty theory was founded by Liu (2007) in 2007 and refined by
Liu (2010a) in 2010. Many researchers have contributed a lot in this area, such as
Gao (2009); You (2009); Peng and Iwamura (2010), and Wang et al. (2012). Nowadays,
uncertainty theory has become a branch of axiomatic mathematics, and has found many
applications in mathematical programming (Liu 2009a; Bhattacharyya et al. 2010),
uncertainriskanalysis(Liu2010c),uncertainfinance(Huang2011),uncertaininference
(Liu 2010b; Gao et al. 2010) and uncertain logic (Liu 2011; Chen and Ralescu 2011).
In the framework of uncertainty theory, Liu (2008) introduced a concept of uncertain
process to study the evolution of uncertain phenomena with time in 2008, and
Liu (2009b) designed a canonical process in 2009. The concept of uncertain integral
was proposed by Liu (2008) to integrate an uncertain process with respect to the
canonical process, and later this type of uncertain integral was called Liu integral by
the academic community. After that, Liu (2009b) recast his work via the fundamental
theorem and thus produced the techniques of chain rule, change of variables, and
integration by parts. Since then, an uncertain calculus theory was founded.
Based on Liu integral, Liu (2008) defined uncertain differential equation driven by
canonical process. Following that, Chen and Liu (2010) gave an existence and uniqueness
theorem for uncertain differential equations. By means of uncertain differential
equation, Liu (2009b) assumed the stock price follows a geometric canonical process,
and proposed an uncertain stock model. Then Chen (2011) gave the American option
pricing formula for the stock model. In 2010, Peng and Yao (2010) proposed another
uncertain stock model to describe the stock prices in long-run. Besides, Zhu (2010)
presented optimal control policy in uncertain environment.
In this paper, we will present a new uncertain calculus which deals with the integration
and differentiation of uncertain process with respect to renewal process instead of
canonical process. The rest of this paper is structured as follows. The next section is
intended to introduce some concepts of uncertain process and Liu integral. In Sect. 3,
uncertain calculus with respect to renewal process is proposed. In Sect. 4, uncertain differential
with respect to renewal process is proposed. In Sect. 5, uncertain differential
equation with jumps is presented. Finally, some remarks are made in Sect. 6.
2 Preliminary
In this section, we will introduce some useful definitions about uncertain process and
uncertain calculus. An uncertain process is a sequence of uncertain variables indexed
123

Uncertain calculus with renewal process 287
by time and space. The most important uncertain processes are canonical process and
renewal process.
Definition 1 (Liu 2009b) An uncertain process Ct is said to be a canonical 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 expected value
0 and variance t 2 , whose uncertainty distribution is
� � ��−1 −π x
�(x) = 1 + exp √ , x ∈ℜ.
3t
Note that �Ct and �t are infinitesimals with the same order. Since Ct is a normal
uncertain variable N(0, t),wehaveCt/tis a normal uncertain variable N(0, 1) by the
operational law of uncertain variables. The uncertain process exp(et + σ Ct) is called
a geometric canonical process, which is usually used to describe the stock price in
uncertain market.
Based on canonical process, an uncertain integral named Liu integral was defined
by Liu (2007), thus offering a theory of uncertain calculus.
Definition 2 (Liu 2009b) LetXtbe an uncertain process and Ct be a canonical
process. For any partition of closed interval [a, b] with a = t1 < t2 < ···< tk+1 = b,
the mesh is written as
Then Liu integral of Xt is defined by
�b
a
� = max
1≤i≤k |ti+1 − ti|.
XtdCt = lim
k�
Xti
�→0
i=1
· (Cti+1 − Cti )
provided that the limit exists almost surely and is finite. For this case, the uncertain
process Xt is said to be Liu integrable.
For example, a continuous function f (t) is Liu integrable, and
�s
0
⎛
�s
⎞
f (t)dCt ∼ ⎝0, | f (t)|dt⎠
is a normal uncertain variable at each time s. The canonical process Ct is a Liu integrable
process, and
�s
0
0
CtdCt = 1
2 C2 s .
123

288 K. Yao
Definition 3 (Liu 2009b) LetCt be a canonical process and Zt be an uncertain
process. If there exist uncertain processes μs and σs such that
Zt = Z0 +
�t
�t
μsds + σsdCs
for any t ≥ 0, then Zt is said to have a Liu differential
0
dZt = μtdt + σtdCt.
For example, the uncertain process C 2 t has a Liu differential dC2 t = 2CtdCt. The
uncertain process tCt has a Liu differential d(tCt) = Ctdt + tdCt.
Liu (2009b) verified the fundamental theorem of uncertain calculus, i.e., for a
canonical process Ct and a continuous differentiable function h(t, c), the uncertain
process Zt = h(t, Ct) is differentiable and has a Liu differential
dZt = ∂h
∂t (t, Ct)dt + ∂h
(t, Ct)dCt.
∂c
Based on the fundamental theorem, Liu proved the chain rule, i.e., for two continuously
differentiable functions f and g, the uncertain process f (g(Ct)) has a Liu
differential
d f (g(Ct)) = f ′ (g(Ct))g ′ (Ct)dCt,
and the integration by parts theorem, i.e., for two Liu differentiable processes Xt and
Yt, the uncertain process XtYt has a Liu differential
d(XtYt) = YtdXt + XtdYt.
Definition 4 (Liu 2008) Letξ1,ξ2,... be iid positive uncertain variables. Define
S0 = 0 and Sn = ξ1 + ξ2 +···+ξn for n ≥ 1. Then the uncertain process
is called a renewal process.
Nt = max
n≥0 {n|Sn ≤ t}
Each sample-path of Nt is a right-continuous and increasing step function taking
only nonnegative integer values. Assuming the interarrival times have a common
uncertainty distribution �, Liu (2010a) proved that Nt has an uncertainty distribution
123
�
�(x) = 1 − �
0
t
⌊x⌋+1
�
,

Uncertain calculus with renewal process 289
where ⌊x⌋ represents the maximal integer less than or equal to x. Besides, Liu (2010a)
proved that Nt/t converges in distribution to 1/ξ1. Based on this, Liu (2010a) proved
the elementary renewal theorem, i.e.,
provided that E[1/ξ1] exists.
lim
t→∞ E
3 Uncertain integral with renewal process
� � � �
Nt 1
= E
t ξ1
Liu integral deals with uncertain calculus with respect to canonical process, which is a
continuous uncertain process. However, emergencies such as war and economic crisis
occur occasionally, and bring the process a sudden shift. To describe such jumps, we
try to found a new kind of uncertain calculus with respect to uncertain renewal process
from this section. First, the concept of uncertain integral with respect to renewal
process is defined as follows.
Definition 5 Let Xt be an uncertain process and Nt a renewal process. For any partition
of a closed interval [a, b] with a = t1 < t2 < ···< tk+1 = b, the mesh is written
as
� = max
i≤i≤k |ti+1 − ti|.
Then the uncertain integral of Xt with respect to Nt is
�b
a
XtdNt = lim
k�
Xti
�→0
i=1
· (Nti+1 − Nti )
provided that the limit exists almost surely and is finite. For this case, the uncertain
process Xt is said to be integrable with respect to Nt.
Example 1 Let Nt be an uncertain renewal process. Then for any partition 0 = t1 <
t2 < ···< tk+1 = s, wehave
That is,
�s
0
dNt = lim
�→0
i=1
k�
(Nti+1 − Nti ) ≡ Ns − N0 = Ns.
�s
0
dNt = Ns. (1)
123

290 K. Yao
Example 2 Let Nt be an uncertain renewal process. Then for any partition 0 = t1 <
t2 < ···< tk+1 = s, wehave
as � → 0. That is,
N 2 s =
=
k�
i=1
k�
i=1
�
N 2 ti+1 − N 2 �
ti
� Nti+1
�
→ Ns + 2
�s
0
0
s
− Nti
NtdNt
� 2 + 2
k�
i=1
Nti
� Nti+1
− Nti
NtdNt = 1
2 Ns(Ns − 1). (2)
Example 3 Let Nt be an uncertain renewal process. Then for any partition 0 = t1 <
t2 < ···< tk+1 = s, wehave
as � → 0. That is,
sNs =
=
k� �
ti+1Nti+1 − ti
�
Nti
i=1
k�
Nti+1 (ti+1 − ti) +
i=1
s
� �s
→ Ntdt + tdNt
0
�s
0
0
k�
i=1
�s
Ntdt + tdNt = sNs
0
�
ti Nti+1
− Nti
Theorem 1 If Nt is a renewal process and Xt is an integrable uncertain process with
respect to Nt on [a, b], then Xt is integrable with respect to Nt on each subinterval
of [a, b]. Moreover, if c ∈[a, b], then
123
�b
a
XtdNt =
�c
a
�
XtdNt +
c
b
XtdNt.
�
�
(3)

Uncertain calculus with renewal process 291
Proof Since Xt is an integrable uncertain process with respect to Nt on [a, b], for any
partition of the closed interval [a, b] with a = t1 < t2 < ···< tk+1 = b, the limit
lim
k�
Xti
�→0
i=1
· (Nti+1 − Nti )
exists almost surely and is finite. Thus for any subinterval [a ′ , b ′ ]⊂[a, b] and any
partition a = t ′ 1 < t′ 2 < ···< t′ k = b, we obtain that the limit
lim
�→0
i=1
k�
Xt ′ · (N
i t ′ − N
i+1 t ′ )
i
exists almost surely and is finite. It follows from Definition 5 that Xt is integrable with
respect to Nt on [a ′ , b ′ ]. Furthermore, for any partition of the closed interval [a, b]
with a = t1 < t2 < ···< tk = c < tk+1 < tk+2 < ···< tn+1 = b, wehave
Thus
n�
i=1
Xti
k−1
�
· (Nti+1 − Nti ) =
�b
a
i=1
Xti
XtdNt =
· (Nti+1 − Nti ) +
�c
a
�
XtdNt +
c
b
n�
i=k+1
XtdNt.
Xti
· (Nti+1 − Nti ).
Theorem 2 If Nt is a renewal process, and Xt and Yt are two integrable uncertain
processes with respect to Nt on [a, b], then αXt + βYt is an integrable uncertain
process with respect to Nt on [a, b] for any real numbers α and β, and
�b
a
�b
�b
(αXt + βYt)dNt = α XtdNt + β YtdNt.
a
Proof Since Xt and Yt are two integrable uncertain processes with respect to Nt on
[a, b], for any partition of the closed interval [a, b] with a = t1 < t2 < ···< tk+1 = b,
the limits
lim
k�
Xti
�→0
i=1
· (Nti+1 − Nti )
a
⊓⊔
123

292 K. Yao
and
lim
k�
Yti
�→0
i=1
exist almost surely and are finite. Thus
�b
a
(αXt + βYt)dNt = lim
�→0
i=1
= α lim
· (Nti+1 − Nti )
k�
(αXti + βYti ) · (Nti+1 − Nti )
k�
Xti
�→0
i=1
a
· (Nti+1 − Nti ) + β lim
�b
�b
= α XtdNt + β YtdNt,
a
k�
Yti
�→0
i=1
· (Nti+1 − Nti )
and the uncertain process αXt + βYt is integrable with respect to Nt on [a, b]. ⊓⊔
4 Uncertain differential with renewal process
Definition 6 Let Nt be a renewal process and Zt an uncertain process. If there exist
uncertain processes μs and γs such that
Zt = Z0 +
�t
�t
μsds + γsdNs
for any t ≥ 0, then Zt is said to have an uncertain differential
0
dZt = μtdt + γtdNt.
For this case, Zt is called a differentiable uncertain process with drift μt and jump γt.
Example 4 Let Nt be an uncertain renewal process. It follows from the Eq. (1) that
Nt =
�t
0
dNs.
Thus the uncertain process Nt is an uncertain differentiable process with respect to
renewal process Nt and has an uncertain differential dNt.
123
0

Uncertain calculus with renewal process 293
Example 5 Let Zt = μt + γ Nt be an uncertain process. Since
Zt =
�t
0
�t
μds + γ dNs,
we obtain that the uncertain process Zt is an uncertain differentiable process with
respect to renewal process Nt and has an uncertain differential
dZt = μdt + γ dNt.
Example 6 Let Nt be an uncertain renewal process. It follows from the Eq. (2) that
N 2 t
�t
�t
= 2 NsdNs + dNs.
0
Thus the uncertain process N 2 t is an uncertain differentiable process with respect to
renewal process Nt and has an uncertain differential
dN 2 t = 2NtdNt + dNt.
Example 7 Let Nt be an uncertain renewal process. It follows from the Eq. (3) that
tNt =
�t
0
0
0
0
�t
Nsds + sdNs.
Thus the uncertain process tNt is an uncertain differentiable process with respect to
renewal process Nt and has an uncertain differential
d(tNt) = Ntdt + tdNt.
Theorem 3 (Fundamental Theorem) Let Nt be a renewal process, and h(t,n) a continuously
differentiable function. Then the uncertain process Zt = h(t, Nt) has an
uncertain differential
dZt = ∂h
∂t (t, Nt)dt + h(t, Nt) − h(t, Nt−).
Proof Since the function h is continuous differentiable, by Taylor series expansion,
we have
�Zt = h(t, Nt) − h(t − �t, Nt−�t)
= h(t, Nt−�t) − h(t − �t, Nt−�t) + h(t, Nt) − h(t, Nt−�t)
= ∂h
∂t (t, Nt−�t)�t + h(t, Nt) − h(t, Nt−�t) + o(�t).
123

294 K. Yao
Letting �t → 0, we have
dh(t, Nt) = ∂h
∂t (t, Nt−)dt + h(t, Nt) − h(t, Nt−)
= ∂h
∂t (t, Nt)dt + h(t, Nt) − h(t, Nt−).
Example 8 Consider the uncertain process μt +γ Nt. For this case, we have h(t, n) =
μt + γ n. It is clear that
∂h
∂t (t, n) = μ, h(t, Nt) − h(t, Nt−) = γ dNt.
It follows from the fundamental theorem that
d(μt + γ Nt) = μdt + γ dNt.
Example 9 Consider the uncertain process tNt. For this case, we have h(t, n) = tn.
It is clear that
∂h
∂t (t, c, n) = n, h(t, Nt) − h(t, Nt−) = tdNt.
It follows from the fundamental theorem that
d(tNt) = Ntdt + tdNt.
Theorem 4 (Integration by Parts Theorem) Suppose Xt and Yt are two differentiable
uncertain processes with respect to renewal process. Then we have
d(XtYt) = YtdXt + XtdYt + (Xt − Xt−)(Yt − Yt−).
Proof For any partition of closed interval of [0, t] with 0 = t1 < t2 < ···< tk+1 = t,
we have
123
XtYt = X0Y0 + lim
�→0
i=1
= X0Y0 + lim
�→0
i=1
k�
(Xti+1
Yti+1 − Xti
Yti )
k�
�
Xti
(Yti+1 − Yti )
+Yti (Xti+1 − Xti ) + (Xti+1 − Xti )(Yti+1 − Yti )� .
⊓⊔

Uncertain calculus with renewal process 295
It follows from the definition of the uncertain integral that
XtYt = X0Y0 +
Thus we have
�
0
t
�
YsdXs +
0
t
XsdYs + �
(Xs − Xs−)(Ys − Ys−).
0

296 K. Yao
Thus the uncertain differential equation has a solution
Xt = X0 + αt + βCt + γ Nt.
Example 12 Let α, β and γ be real numbers. Consider an uncertain differential equation
with jumps
It is equivalent to
Integrating on both sides, we have
�t
0
dXt = αXtdt + β XtdCt + γ Xt N −1
t dNt.
X −1
t dXt = αdt + βdCt + γ N −1
t dNt.
X −1
s dXs =
�t
0
�t
�
αds + βdCs +
Thus the uncertain differential equation has a solution
0
Xt = X0 exp(αt + βCt) · γ Nt.
0
t
γ N −1
s dNs.
Example 13 Let α, β and γ be real numbers. Consider an uncertain differential equation
with jumps
dXt = αXtdt + β XtdCt + γ Xt−dNt.
It is easy to verify that the uncertain differential equation has a solution
6 Conclusions
Xt = X0 exp(αt + βCt)(1 + γ) Nt .
This paper first proposed an uncertain calculus with respect to renewal process to
supplement the uncertain calculus theory. Then it gave the fundamental theorem of
uncertain calculus with renewal process. Besides, this paper presented an uncertain
differential equation with jumps to model the sudden drifts in uncertain systems.
Acknowledgments This work was supported by National Natural Science Foundation of China Grant
No. 91024032 and No. 60874067.
123

Uncertain calculus with renewal process 297
References
Bhattacharyya, R., Chatterjee, A., & Kar, S. (2010). Uncertainty theory based novel multi-objective
optimization technique using embedding theorem with application to R& D project portfolio
selection. Applied Mathematics, 1, 189–199.
Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political
Economy, 81, 637–654.
Chen, X. (2011). American option pricing formula for uncertain financial market. International Journal
of Operations Research, 8(2), 32–37.
Chen, X., & Liu, B. (2010). Existence and uniqueness theorem for uncertain differential equations. Fuzzy
Optimization and Decision Making, 9(1), 69–81.
Chen, X., & Ralescu, D. A. (2011). A note on truth value in uncertain logic. Expert Systems with
Applications, 38, 15582–15586.
Gao, X. (2009). Some properties of continuous uncertain measure. International Journal of Uncertainty,
Fuzziness and Knowledge-Based Systems, 17(3), 419–426.
Gao, X., Gao, Y., & Ralescu, D. A. (2010). On Liu’s inference rule for uncertain systems. International
Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 18(1), 1–11.
Huang, X. (2011). Mean-risk model for uncertain portfolio selection. Fuzzy Optimization and Decision
Making, 10(1), 71–89.
Ito, K. (1944). Stochastic integral (pp. 519–524). Tokyo, Japan: Proceedings of the Japan Academy.
Kalman, R. E., & Bucy, R. S. (1961). New results in linear filtering and prediction theory. Journal of
Basic Engineering, 83, 95–108.
Liu, B. (2007). Uncertainty Theory (2nd ed.). Berlin: Springer.
Liu, B. (2008). Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems, 2(1),
3–16.
Liu, B. (2009a). Theory and practice of uncertain programming (2nd ed.). Berlin: Springer.
Liu, B. (2009b). Some research problems in uncertainty theory. Journal of Uncertain Systems, 3(1), 3–10.
Liu, B. (2010a). Uncertainty theory: A branch of mathematics for modeling human uncertainty.
Berlin: Springer.
Liu, B. (2010b). Uncertain set theory and uncertain inference rule with application to uncertain
control. Journal of Uncertain Systems, 4(2), 83–98.
Liu, B. (2010c). Uncertain risk analysis and uncertain reliability analysis. Journal of Uncertain Systems,
4(3), 163–170.
Liu, B. (2011). Uncertain logic for modeling human language. Journal of Uncertain Systems, 5(1), 3–20.
Peng, J., & Yao, K. (2010). A new option pricing model for stocks in uncertainty markets. International
Journal of Operations Research, 7(4), 213–224.
Peng, Z., & Iwamura, K. (2010). A sufficient and necessary condition of uncertainty distribution. Journal
of Interdisciplinary Mathematics, 13(3), 277–285.
Wang, X., Gao, Z., & Guo, H. (2012). Delphi method for estimating uncertainty distributions, Information:
An International Interdisciplinary Journal, 15(2), 449–460.
Wiener, N. (1923). Differential space. Journal of Mathematical Physics, 2, 131–174.
You, C. (2009). Some convergence theorems of uncertain sequences. Mathematical and Computer
Modelling, 49(3–4), 482–487.
Zhu, Y. (2010). Uncertain optimal control with application to a portfolio selection model. Cybernetics
and Systems, 41(7), 535–547.
123