Liu process and uncertain calculus
© Chen and Ralescu; licensee Springer. 2013
Received: 16 February 2013
Accepted: 19 April 2013
Published: 19 June 2013
Uncertain calculus is a branch of mathematics that deals with the integral and differential of functions of uncertain processes. This paper first introduces the Liu process as an uncertain process defined by the Liu integral. Some properties of Liu processes are investigated such as sample continuity property, finite variation property, and the fact that a continuously differentiable function of the Liu process is another Liu process, among others. Based on the Liu process, the uncertain integral is extended. Furthermore, some mathematical properties are proved, including the fundamental theorem, change of variable theorem, and integration by parts theorem. Finally, uncertain calculus with respect to multiple Liu processes is discussed.
A stochastic process, a sequence of random variables indexed by time, is a useful tool to deal with dynamical random phenomena. A very important such process is called the Wiener process and was defined by Wiener . The Wiener process, also known as the Brownian motion, is a stochastic process with stationary independent increments, and the increments are normal random variables. Based on the Wiener process, stochastic calculus was developed by Ito . It is a branch of mathematical theory dealing with the integration and differentiation of functions of a stochastic process. Stochastic calculus with respect to the Wiener process is also called the Ito calculus. It has important applications in asset pricing theory. In 1967, the stochastic integral was extended by Kunita and Watanabe  to square integrable martingales. Furthermore, stochastic integral with respect to a semimartingale was introduced by Doléans-Dade and Meyer . Besides, stochastic calculus with respect to a local martingale, the Poisson process, and the Lévy process have been studied (see [5, 6]).
Stochastic processes are defined based on probability theory. When we use it, a large sample size is needed to estimate probability distribution based on long-run frequency. However, Liu  pointed out that the sample size is often too small (even no sample) in practice and the degree of belief usually has much larger variance than the long-run frequency. Thus, we should deal with it by using uncertainty theory. These facts promoted Liu  to found an uncertainty theory. That is a branch of mathematics dealing with human uncertainty. In order to describe dynamic uncertain systems, an uncertain process was introduced by Liu  as a sequence of uncertain variables indexed by time. In addition, Zhang et al.  proposed a delayed renewal process. Chen  investigated some properties of uncertain stationary independent increments.
Based on the Liu canonical process, a type of uncertain process with stationary independent increments which follow normal uncertainty distribution, Liu  invented an uncertain calculus in 2009. This type of uncertain integral is called the Liu integral. It was extended to multiple Liu canonical processes . It is used to deal with the integration and differentiation of uncertain processes. The theory of integration and differentiation of uncertain processes with respect to the Liu process is called the Liu calculus. 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 . We call this type of uncertain integral the Yao integral. The theory of integration and differentiation of uncertain processes with respect to a renewal process is called the Yao calculus. Chen  introduced an uncertain integral with respect to a finite variation process.
In addition, the uncertain differential equation driven by the Liu canonical process was introduced by Liu . After that, Chen and Liu  proved the existence and uniqueness theorem for uncertain differential equations. Yao and Gao  studied stability theorems for uncertain differential equations. Chen and Liu  proposed solution methods for linear uncertain differential equations. Liu  and Yao  gave a method to solve nonlinear uncertain differential equations. Besides, Yao and Chen  introduced a numerical method for uncertain differential equations. Meanwhile, an uncertain differential equation has been applied to uncertain optimal control by Zhu , American option pricing by Chen , and other option pricing models by Peng and Yao . Liu  discussed some basic concepts of uncertain finance. For the latest development of uncertainty theory, please see .
In this paper, we generalize the Liu process by the Liu integral. Our goal is to extend an uncertain integral with respect to the Liu process. This uncertain integral has the properties of continuity and linearity. In the framework of the uncertain integral, the fundamental theorem of differentiation of function of uncertain processes is proved. In addition, the integration by parts formula and the formula for change of variables are derived. The rest of the paper is organized as follows: some preliminary concepts of the uncertain process including the Liu calculus, Yao calculus, and uncertain calculus with respect to an uncertain finite variation process are recalled in the ‘Preliminary’ section. The uncertain integral with respect to the general Liu process will be defined in the ‘Liu process’ section. The uncertain differential will be discussed in the ‘Uncertain integral with respect to the Liu process’ section. At last, a brief summary is given in the ‘Multifactor Liu process’ section.
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.
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.
provided that at least one of the two integrals is finite.
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].
is an event.
is called a renewal process.
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  proved that the expected value of stationary independent increment process S t is E[S t ] = a + bt.
( ) 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.
In order to deal with the integration and differentiation of uncertain processes, Liu  proposed an uncertain integral with respect to the Liu process.
provided that the limit exists almost surely and is finite. In this case, the uncertain process X t is said to be Liu integrable.
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 .
provided that the limit exists almost surely and is finite. In this case, the uncertain process X t is said to be Yao integrable.
Uncertain integral with finite variation processes
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 .
Uncertain differential equation
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.
for some constant L. Moreover, the solution is sample-continuous.
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  studied the integral of solution to uncertain differential equations.
A canonical Liu process C t is a Liu process with initial value 0, drift 0, and diffusion 1.
An arithmetic Liu process X t = e t + σ C t is a Liu process with initial value 0, drift e, and diffusion σ.
A geometric Liu process X t = exp(e t + σ C t ) is a Liu process with initial value 1, drift e X t , and diffusion σ X t .
The uncertain process is a Liu process with initial value 0, drift 2t, and diffusion .
is a Liu process with drift f(t,X t ) and diffusion g(t,X t ).
Let h(t,c) be a continuously differentiable function. Then h(t,C t ) is a Liu process with drift and diffusion .
The theorem is proved. Therefore, a Liu process is an uncertain finite variation process. □
The Liu process is a sample-continuous uncertain process.
is finite for almost all the sample path γ ∈ Γ. It follows from the continuity of the integral that and as Δt → 0 almost surely. Therefore, the theorem is proved. □
Uncertain integral with respect to the Liu process
The uncertain integral of an uncertain process with respect to the Liu process is also a Liu process.
It follows from the definition of the Liu process that Z t is indeed a Liu process. □
Hence, the theorem is proved. □
The theorem is proved. □
Multifactor Liu process
This paper introduced the concept of the Liu process which is defined by the Liu integral. Based on the Liu process, we extended the Liu integral on such a process. Some basic properties of this integral were discussed. Furthermore, the uncertain differential was introduced, and the fundamental theorem of uncertain calculus was derived. The integration by parts theorem was also discussed. Finally, we have studied multifactor Liu processes.
This work was supported by the National Natural Science Foundation of China (grant no. 61273044), Nankai University Project Funds for Young Teachers (no. NKQ1118), and Tianjin Municipal Research Program of Application Foundation and Advanced Technology of China (grant no. 10JCYBJC07300).
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