Estimating the Variance of the Square of Canonical Process

Transcription

1 Estimating the Variance of the Square of Canonical Process Youlei Xu Uncertainty Theory Laboratory, Department of Mathematical Sciences Tsinghua University, Beijing 184, China Abstract Canonical process, as an important kind of uncertain process, has been applied to uncertain calculus, uncertain differential equation and uncertain finance. This paper gives a method to estimate the variance of the square of canonical process. Keywords: uncertain process, canonical process, epected value, variance 1 Introduction Some information and knowledge represented by human language are usually in the states of uncertainty. Perhaps some people use randomness or fuzziness to describe that. However, a lot of surveys showed that the imprecise quantities represented by human language behave neither like randomness nor like fuzziness. In order to model those imprecise quantities, Liu [1] founded uncertainty theory in 7 and refined [] it in 1. Nowadays uncertainty theory has become a branch of mathematics based on normality, self-duality, countable subadditivity, and product measure aioms. In order to collect and interpret epert s eperimental data by uncertainty theory, Liu [] introduced a questionnaire survey and proposed uncertain statistics in 1. As an application of uncertainty theory, Liu [] proposed uncertain programming which is a type of mathematical programming involving uncertain variables. Besides, Liu [4] introduced uncertain risk analysis and reliability analysis as tools to deal with system risk and reliability via uncertainty theory. Moreover, uncertain set theory was proposed by Liu [5] as a generalization of uncertainy theory to the domain of uncertain sets. Then Liu [6] designed uncertain logic to deal with uncertain knowledge via uncertain set theory. In order to derive consequences from uncertain knowledge or evidence, Liu [5] introduced uncertain inference and proposed the first inference rule, then Gao, Gao and Ralescu [7] etended the inference rule to the case with multiple antecedents and with multiple if-then rules. With the development of uncertainty theory, Liu [8] introduced a concept of uncertain process that is 1

2 essentially a sequence of uncertain variables indeed by time and space, and Liu [9] designed a canonical process, as an important kind of uncertain process, which is used to describe the irregular movement of pollen with finite speed. In addition, Liu [9] proposed uncertain calculus to deal with differentiation and integration of function of uncertain processes, and Liu [8] defined uncertain differential equation. After that, Chen and Liu [1] proved an eistence and uniqueness theorem for uncertain differential equations. By means of uncertain differential equation, Liu [9] proposed an uncertain stock model named Liu s stock model and derived European option pricing formulas. Following that, numbers of researchers have continued in uncertain finance. In this paper, we will give a method to estimate the variance of the square of canonical process. The rest of the paper is organized as follows. Some concepts of uncertain process and variance are recalled in section. We will estimate the variance of the square of canonical process in section. A brief summary is given in section 4. Preliminary An uncertain process is essentially a sequence of uncertain variables indeed by time or space. The study of uncertain process was started by Liu [8] in 8. Definition 1. (Liu [8]) Let T be an inde set and let (Γ, L, M) be an uncertainty space. An uncertain process is a measurable function from T (Γ, 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 Γ X t (γ) B} is an event. Definition. (Liu [9]) An uncertain process C t is said to be canonical process if (i) C and almost all sample paths are Lipschitz continuous, (ii) C t has stationary and independent increments, (iii) every increment C s+t C s is a normal uncertain variable with epected value and variance t. Epected value is the average value of uncertain variable in the sense of uncertain measure, and represents the size of uncertain variable. Definition. (Liu [1]) Let ξ be an uncertain variable. Then the epected value of ξ is defined by E[ξ] provided that at least one of the two integrals is finite. M{ξ r}dr M{ξ r}dr

3 The variance of uncertain variable, as an important criteria for ranking uncertain variable, provides a degree of the spread of the distribution around its epected value. Definition 4. (Liu [1]) Let ξ be an uncertain variable with finite epected value e. Then the variance of ξ is defined by V [ξ] E[(ξ e) ]. Theorem 1. Let C t be a canonical process. Then C t is an uncertain variable with epected value and variance t, and has an uncertainty distribution Φ() ( ( 1 + ep π )) 1 (1) at each time t >. That is, C t N(, t). Theorem. Let C t be a canonical process. Then C t is an uncertain variable and.5t E[C t ] t. Estimating the Variance of the Square of Canonical Process In this section, we will estimate the variance of the square of canonical process. Theorem. Let C t be a canonical process. Then C t is an uncertain variable and 1.6t 4 < V [C t ] < 4.1t 4. () Proof: Note that C t is a normal uncertain variable and has an uncertainty distribution (1) with finite epected e. It follows from the definition of variance that V [C t ] M{(C t e) }d M{(C t e + ) (C t e + ) ( e C t e )}d. On one hand, we have V [C t ] e + } + M{C t e + } + M{ e C t e )}d.

4 Since.5t e t, and Thus, we have e + }d t e + }d.5t + }d 1 Φ(.5t + ( )d y.5t + ) 4y yt 4y yt 4y yt + 1.6t t t 4, M{ e C t e )}d < < t t 4y yt yt y dy M{C t e + }d 1.75t 4, M{C t e )}d Φ( t ( )d y t ) 4yt 4y 1 + ep( πy 4yt 4y 1 + ep( π t ep( π ) <.86t4. V [C t ] < 1.75t t t 4 4.1t 4. On the other hand, we have V [C t ] > t e + }d 1 Φ( t + ( )d y t + ) 4y 4yt 4y 4yt 1 + ep( πy 4y 4yt 1.1t 4 +.5t 4 1.6t 4. 4y 4yt + yt y dy The theorem is verified. 4

5 4 Conclusions This paper gives an estimate of the variance of Ct. As a result of theorem and theorem, we have that Ct and t are of same order, which is very useful in uncertain calculus and uncertain differential equation. Acknowledgements This work was supported by National Natural Science Foundation of China Grant No References [1] Liu B., Uncertainty Theory, nd ed., Springer-Verlag, Berlin, 7. [] Liu B., Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 1. [] Liu B., Theory and Practice of Uncertain Programming, nd ed., Springer-Verlag, Berlin, 9. [4] Liu B., Uncertain risk analysis and uncertain reliability analysis, Journal of Uncertain Systems, Vol.4, No., 16-17, 1. [5] Liu B., Uncertain set theory and uncertain inference rule with application to uncertain control, Journal of Uncertain Systems, Vol.4, No., 8-98, 1. [6] Liu B., Uncertain logic for modelling human language, Journal of Uncertain Systems, Vol.5, No.1, -, 11. [7] Gao X., Gao Y. and Ralescu D.A., On Liu s inference rule for uncertain systems, International Journal of Uncertainy, Fuzzy and Knowledge-Based Systems, Vol.18, No.1, 1-11, 1. [8] Liu B., Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, Vol., No.1, -16, 8. [9] Liu B., Some research problems in uncertainty theory, Journal of Uncertain Systems, Vol., No.1, -1, 9. [1] Chen X. and Liu B., Eistence and uniqueness theorem for uncertain differential equations, Fuzzy Optimization and Decision Making, Vol.9, No.1, 69-81, 1. 5