Tail Value-at-Risk in Uncertain Random Environment

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1 Noname manuscript No. (will be inserted by the editor) Tail Value-at-Risk in Uncertain Random Environment Yuhan Liu Dan A. Ralescu Chen Xiao Waichon Lio Abstract Chance theory is a rational tool to be used in the systems which contain not only uncertainty but also randomness. In this paper, the concept of tail value-atrisk in uncertain random risk analysis is proposed and some theorems are provided for its calculation. Moreover, the tail value-at-risk is applied as the right-tail in the parallel system, series system, standby system, k-out-of-n system and structural system. Keywords Risk Analysis Uncertainty Theory Chance Theory Tail Value-at-Risk 1 Introduction Random phenomena is common in the daily life, and the risk often comes with the randomness. To handle the risk in random environment, Roy [23] proposed probabilistic risk analysis by the risk index (safety-first criterion) which is essentially the probability measure for the loss. Given normal market conditions, the concept of value-at-risk (VaR) was further suggested by the leading bank Morgan [2] to calculate the maximal possible loss for reporting firm-wide risk. After that, tail value-at-risk (TVaR), also known as tail conditional expectation, was presented by Artzner et al. [2] as a measure which is more coherent and conservative than the Yuhan Liu Dan A. Ralescu Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH , USA liu3yh@mail.uc.edu Chen Xiao School of Finance, Nankai University, Tianjin 335, China xc@mail.nankai.edu.cn * Corresponding author: Waichon Lio Department of Mathematical Sciences, Tsinghua University, Beijing, 184, China wj-liao16@mails.tsinghua.edu.cn value-at-risk measurement, and the expected shortfall (ES) was proposed by Acerbi and Tasche [1] as another coherent risk measure. Probabilistic risk analysis has been successfully applied in many fields, including engineering and management science. However, it should be noted that the fundamental assumption for obtaining the probability distribution is that there are adequate historical data. In real life, it is impossible to collect enough samples for estimating the probability distribution in many cases. Therefore, we have to invite some domain experts to give out the belief degrees that each event may happen, but it is common for human beings overweighting unlikely events (Kahneman and Tversky [5]). Furthermore, Liu [13] pointed out that human beings usually estimate a much wider range of values than the object actually takes, and Liu [11] demonstrated that some counterintuitive results are caused if probability theory is applied to model the belief degrees. To rationally model belief degrees, uncertainty theory was founded by Liu [6]. Since then, uncertainty theory has been developed by many scholars. For instance, Liu [7] introduced uncertain programming and applied it to the machine scheduling problem, vehicle routing problem and project scheduling problem. Besides, Liu [1] proposed a reliability index for the uncertain systems based on uncertainty theory. Meanwhile, Liu [1] suggested the concept of risk index which is essentially the uncertain measure for some specific loss. According to the framework of uncertain risk analysis, Peng [21] developed a methodology of uncertain value-at-risk and tail valueat-risk. In many cases, a system is complex and contains not only uncertainty but also randomness. To describe these phenomena, Liu [15] founded chance theory by the concepts of chance measure and uncertain random variable, as well as Liu [16] introduced uncertain random

2 2 Yuhan Liu et al. programming based on chance theory. For more applications of chance theory in decision problems, Zhou et al. [24] proposed the methodology of multi-objective optimization in the uncertain random situations. Liu [12] presented uncertain random networks and uncertain random graphs. Ke et al. [4] introduced uncertain random multilevel programming and extended its application to the product control problem, and Qin [22] introduced uncertain random goal programming. Furthermore, a reasonable explanation for the Ellsberg experiment was given by Liu [14] in terms of chance theory. In the uncertain random environments, Liu and Ralescu [17] introduced risk index and developed uncertain random risk analysis. Based on this framework, Liu and Ralescu [18][19] presented a tool of value-at-risk and a concept of expected loss to measure the risk in the uncertain random system. This paper will extend the value-at-risk to the tool of tail value-at-risk. Then some theorems will be verified to evaluate the tail value-atrisk. Finally, we will apply the concept of tail valueat-risk to the parallel system, series system, standby system, k-out-of-n system and structural system. 2 Preliminaries Uncertainty theory is an axiomatic mathematical branch to model the belief degrees, and chance theory is a methodology to model a complex system which contains not only uncertainty but also randomness. This section introduces some basic concepts and theorems of uncertainty theory and chance theory. 2.1 Uncertainty Theory Let Γ be a nonempty set, and let L be a σ-algebra over Γ. For any element Λ in L is said to be an event. Liu [6] defined a real-valued set function which is called an uncertain measure by the following three axioms: Axiom 1. (Normality Axiom) M{Γ } 1 for the universal set Γ. Axiom 2. (Duality Axiom) M{Λ} + M{Λ c } 1 for any event Λ. Axiom 3. (Subadditivity Axiom) For every countable sequence of events Λ 1, Λ 2,, we have { } M Λ i M{Λ i }. i1 i1 Then the triplet (Γ, L, M) is called an uncertainty space. Moreover, Liu [8] defined a product uncertain measure on the product σ-algebra L by the fourth axiom as follows: Axiom 4. (Product Axiom) Let (Γ k, L k, M k ) be uncertainty spaces for k 1, 2,. The product uncertain measure M is an uncertain measure satisfying { } M Λ k M k {Λ k } k1 k1 where Λ k are arbitrarily chosen events from L k for k 1, 2,, respectively. An uncertain variable was defined by Liu [6] as a measurable function from an uncertainty space (Γ, L, M) to the set of real numbers such that for any Borel set B of real numbers, the set {ξ B} {γ Γ ξ(γ) B} is an event. Meanwhile, Liu [6] defined the uncertainty distribution as Φ(x) M{ξ x}, x R to describe the uncertain variable ξ. The uncertainty distribution Φ(x) is called regular if it is a continuous and strictly increasing function with respect to x at which < Φ(x) < 1, as well as satisfies lim and lim x Φ(x) x Φ(x) 1. (Liu [9]). Suppose ξ is an uncertain variable with regular uncertainty distribution Φ(x). Then the inverse function Φ 1 (α) is called the inverse uncertainty distribution of ξ (Liu [9]). The independence of uncertain variables was proposed by Liu [8]. The uncertain variables ξ 1, ξ 2,, ξ n are called independent if for any Borel sets B 1, B 2,, B n of real numbers, we have { n } M (ξ i B i ) i1 n M {ξ i B i }. i1 Now suppose ξ 1, ξ 2,, ξ n are independent uncertain variables which have uncertainty distributions Φ 1, Φ 2,, Φ n, respectively, and 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, m n. Liu [9] proved that the inverse uncertainty distribution of ξ f(ξ 1, ξ 2,, ξ n ) is Φ 1 (α) f(φ 1 1 (α),, Φ 1 m (α), Φ 1 m+1 (1 α),, Φ 1(1 α)). n (1) To measure the size of uncertain variable ξ, the concept of expected value was suggested by Liu [6] as E[ξ] + M{ξ x}dx M{ξ x}dx

3 Tail Value-at-Risk in Uncertain Random Environment 3 provided that at least one of the two integrals is finite. Furthermore, if ξ is an uncertain variable with regular uncertainty distribution Φ, then Liu [9] showed that E[ξ] Chance Theory Φ 1 (α)dα. (2) Let (Γ, L, M) be an uncertainty space and (Ω, A, Pr) be a probability space. The product (Γ, L, M) (Ω, A, Pr) is called a chance space. Essentially, it is just another triplet, (Γ Ω, L A, M Pr) (3) where Γ Ω is the universal set, L A is the product σ-algebra, and M Pr is the product measure. Definition 1 (Liu [15]) Suppose (Γ, L, M) (Ω, A, Pr) is a chance space, and Θ L A is an event. Then the chance measure of Θ is defined as Ch{Θ} 1 Pr {ω Ω M{γ Γ (γ, ω) Θ} r} dr. (4) It has been proved by Liu [15] that the chance measure Ch{Θ} is monotone increasing with respect to Θ, as well as satisfies Ch{Γ Ω} 1 (5) and Ch{Θ} + Ch{Θ c } 1 (6) for any event Θ L A. Besides, Hou [3] showed that the chance measure is subadditive, i.e., for any countable sequence of events Θ 1, Θ 2,, we have { } Ch Θ i Ch{Θ i }. (7) i1 i1 Definition 2 (Liu [15]) An uncertain random variable is a measurable function ξ from a chance space (Γ, L, M) (Ω, A, Pr) to the set of real numbers, i.e., {ξ B} is an event for any Borel set B of real numbers. The chance distribution of an uncertain random variable ξ is defined as Φ(x) Ch{ξ x} for any x R. If the chance distribution Φ is a continuous function, then for any x R, we have Ch{ξ x} Φ(x), Ch{ξ x} 1 Φ(x). (8) Theorem 1 (Liu [16]) Suppose η 1, η 2,, η m are independent random variables with probability distributions Ψ 1, Ψ 2,, Ψ m, and τ 1, τ 2,, τ n are independent uncertain variables with regular uncertainty distributions Υ 1, Υ 2, Υ n, respectively. If the function f(η 1,, η m, τ 1,, τ n ) is further assumed to be strictly increasing with respect to τ 1,, τ k and strictly decreasing with respect to τ k+1,, τ n, then the uncertain random variable ξ f(η 1,, η m, τ 1,, τ n ) has a chance distribution Φ(x) g(x; y 1,, y m )dψ 1 (y 1 ) dψ m (y m ) R m (9) where g(x; y 1,, y m ) is the root α of the equation f( y 1,, y m, Υ1 1 (α),, (α), 1 k+1 (1 α),, Υn (1 α)) x. Definition 3 (Liu [15]) Suppose ξ is an uncertain random variable. Then its expected value is E[ξ] + Ch{ξ x}dx Ch{ξ x}dx (1) provided that at least one of the two integrals is finite. 3 Tail Value-at-Risk In 214, the risk index was proposed by Liu and Ralescu [17] to handle the complicated systems which contain uncertainty and randomness. In some cases, we have to know the scale of loss in the uncertain random environment. Therefore, Liu and Ralescue [18] gave out the concept of value-at-risk as follows: Definition 4 (Liu [18]) Suppose a system has uncertain random factors ξ 1, ξ 2,, ξ n and a loss function f. Then the value-at-risk is VaR(α) sup{x Ch{f(ξ 1, ξ 2,, ξ n ) x} α} for any given confidence level α (,1]. A shortage of value-at-risk is that it cannot give any information for the severity of loss. For instance, a loss function with thicker tail is more risky than the other functions, but the value-at-risk may show that they are equal at varied confidence levels. Taking this point of view, we define the tail value-at-risk to deal with this problem. k

4 4 Yuhan Liu et al. Definition 5 Suppose a system has uncertain random factors ξ 1, ξ 2,, ξ n and a loss function f. Then the tail value-at-risk (TVaR) is VaR(β)dβ for any given confidence level α (,1]. In order to calculate the tail value-at-risk, some theorems are proved below based on the ordinary assumptions. Theorem 2 Suppose a system has uncertain random factors ξ 1, ξ 2,, ξ n, and a loss function f. If f(ξ 1, ξ 2,, ξ n ) is further supposed to have a continuous chance distribution Φ(x), then for any confidence level α (,1], the tail value-at-risk satisfies sup{x Φ(x) 1 β}dβ. Proof It can be obtained from Definition 5, equation (6), and equation (8) that sup {x Ch{f(ξ 1, ξ 2,, ξ n ) x} β}dβ sup {x 1 Ch{f(ξ 1, ξ 2,, ξ n ) < x} β}dβ sup {x 1 Φ(x) β}dβ sup {x Φ(x) 1 β}dβ. Theorem 3 Suppose a system has uncertain random factors ξ 1, ξ 2,, ξ n, and a loss function f. If f(ξ 1, ξ 2,, ξ n ) is further supposed to have a regular chance distribution Φ(x), then for any confidence level α (,1], the tail value-at-risk satisfies Proof From Theorem 2, we obtain Φ 1 (1 β)dβ. sup{x Φ(x) 1 β}dβ sup{x x Φ 1 (1 β)}dβ Φ 1 (1 β)dβ. Theorem 4 Suppose a system has independent random variables η 1, η 2,, η m with probability distributions Ψ 1, Ψ 2,, Ψ m and independent uncertain variables τ 1, τ 2,, τ n with regular uncertainty distributions Υ 1, Υ 2,, Υ n, respectively, and its loss function f(η 1, η 2,, η m, τ 1, τ 2,, τ n ) is a strictly increasing function with respect to τ 1,, τ k and a strictly decreasing function with respect to τ k+1,, τ n. Then sup { x R m g(x; y 1,, y m )dψ 1 (y 1 ) dψ m (y m ) 1 β} dβ f( y 1, y 2,, y m, Υ1 1 (α),, (α), 1 k+1 (1 α),, Υn (1 α)) x. Proof It follows from Theorem 1 that f(η 1, η 2,, η m, τ 1, τ 2,, τ n ) has a continuous chance distribution Φ(x) g(x; y 1,, y m )dψ 1 (y 1 ) dψ m (y m ) R m f( y 1, y 2,, y m, Υ1 1 (α),, (α), 1 k+1 (1 α),, Υn (1 α)) x. Then from Theorem 2, we immediately obtain 4 Parallel System sup { x R m g(x; y 1,, y m )dψ 1 (y 1 ) dψ m (y m ) 1 β} dβ. Theorem 5 Considering a parallel system in which there are m elements whose lifetimes are independent random variables η 1, η 2,, η m with continuous probability distributions Ψ 1, Ψ 2,, Ψ m and n elements whose lifetimes are independent uncertain variables τ 1, τ 2,, τ n with continuous uncertainty distributions Υ 1, Υ 2,, Υ n, respectively. Suppose the loss is realized in case the system fails before time T. Then sup {x (Υ 1 (T x) Υ n (T x)) Ψ 1 (T x) Ψ m (T x) β}dβ. Proof Since the lifetime of the parallel system is η 1 η 2 η m τ 1 τ 2 τ n, the loss function f(η 1, η 2,, η m, τ 1, τ 2,, τ n ) is T η 1 η 2 η m τ 1 τ 2 τ n. k k

5 Tail Value-at-Risk in Uncertain Random Environment 5 And the chance distribution of f is Φ(x) Ch{T η 1 η 2 η m τ 1 τ 2 τ n x} Ch{η 1 η 2 η m τ 1 τ 2 τ n T x} 1 Pr {ω Ω M{η 1 (ω) η m (ω) τ 1 τ n T x} r}dr M{y 1 y m τ 1 τ n T x} R m dψ 1 (y 1 ) dψ m (y m ) M {y 1 y m τ 1 τ n y 1 y m<t x T x}dψ 1 (y 1 ) dψ m (y m ) + M {y 1 y m τ 1 τ n y 1 y m T x T x}dψ 1 (y 1 ) dψ m (y m ) M{τ 1 τ 2 τ n T x} y 1 y m<t x dψ 1 (y 1 ) dψ m (y m ) + 1 dψ 1 (y 1 ) dψ m (y m ) y 1 y m T x (1 Υ 1 (T x) Υ n (T x))ψ 1 (T x) Ψ m (T x) + 1 Ψ 1 (T x) Ψ m (T x) 1 (Υ 1 (T x) Υ n (T x))ψ 1 (T x) Ψ m (T x). sup {x 1 (Υ 1 (T x) Υ n (T x)) Ψ 1 (T x) Ψ m (T x) 1 β}dβ sup {x (Υ 1 (T x) Υ n (T x)) The theorem is thus proved. 5 Series System Ψ 1 (T x) Ψ m (T x) β}dβ. Theorem 6 Considering a series system in which there are m elements whose lifetimes are independent random variables η 1, η 2,, η m with continuous probability distributions Ψ 1, Ψ 2,, Ψ m and n elements whose lifetimes are independent uncertain variables τ 1, τ 2,, τ n with continuous uncertainty distributions Υ 1, Υ 2,, Υ n, respectively. Suppose the loss is realized in case the system fails before time T. Then sup {x (1 Υ 1 (T x) Υ n (T x))(1 Ψ 1 (T x)) (1 Ψ m (T x)) 1 β}dβ. Proof Since the lifetime of the series system is η 1 η 2 η m τ 1 τ 2 τ n, the loss function f(η 1, η 2,, η m, τ 1, τ 2,, τ n ) is T η 1 η 2 η m τ 1 τ 2 τ n. And the chance distribution of f is Φ(x) Ch{T η 1 η 2 η m τ 1 τ 2 τ n x} Ch{η 1 η 2 η m τ 1 τ 2 τ n T x} 1 Pr {ω Ω M{η 1 (ω) η m (ω) τ 1 τ n T x} r}dr M {y 1 y m τ 1 τ n R m T x}dψ 1 (y 1 ) dψ m (y m ) M {y 1 y m τ 1 τ n y 1 y m<t x T x}dψ 1 (y 1 ) dψ m (y m ) + M {y 1 y m τ 1 τ n y 1 y m T x T x}dψ 1 (y 1 ) dψ m (y m ) (1 M{τ 1 τ n < T x}) y 1 y m T x dψ 1 (y 1 ) dψ m (y m ) (1 Υ 1 (T x) Υ n (T x)) (1 Ψ 1 (T x)) (1 Ψ m (T x)). The theorem is thus proved. 6 Standby System sup {x (1 Υ 1 (T x) Υ n (T x))(1 Ψ 1 (T x)) (1 Ψ m (T x)) 1 β}dβ. Theorem 7 Considering a standby system in which there are m elements whose lifetimes are independent random variables η 1, η 2,, η m with probability distributions Ψ 1, Ψ 2,, Ψ m and n elements whose lifetimes

6 6 Yuhan Liu et al. are independent uncertain variables τ 1, τ 2,, τ n with regular uncertainty distributions Υ 1, Υ 2,, Υ n, respectively. Suppose the loss is realized in case the system fails before time T. Then sup { x R m g(x; y 1,, y m )dψ 1 (y 1 ) dψ m (y m ) 1 β} dβ 1 (1 α) + 2 (1 α) + + n (1 α) T (y 1 + y y m + x). Proof Since the lifetime of the standby system is η 1 + η η m + τ 1 + τ τ n, the loss function f(η 1, η 2,, η m, τ 1, τ 2,, τ n ) is T (η 1 + η η m + τ 1 + τ τ n ). It follows from Theorem 1 that the chance distribution of f is Φ(x) Ch {T (η 1 + η η m +τ 1 + τ τ n ) x} Ch {η 1 + η η m +τ 1 + τ τ n T x} g(x; y 1,, y m )dψ 1 (y 1 ) dψ m (y m ) R m 1 (1 α) + 2 (1 α) + + n (1 α) T (y 1 + y y m + x). The theorem is thus proved. 7 k-out-of-(m + n) System sup { x R m g(x; y 1,, y m )dψ 1 (y 1 ) dψ m (y m ) 1 β} dβ. Theorem 8 Considering a k-out-of-(m + n) system in which there are m elements whose lifetimes are independent random variables η 1, η 2,, η m with probability distributions Ψ 1, Ψ 2,, Ψ m and n elements whose lifetimes are independent uncertain variables τ 1, τ 2,, τ n with regular uncertainty distributions Υ 1, Υ 2,, Υ n, respectively. Suppose the loss is realized in case the system fails before time T. Then sup { x R m g(x; y 1,, y m )dψ 1 (y 1 ) dψ m (y m ) 1 β} dβ k-max {y 1, y 2,, y m, Υ1 1 (1 α), Υ2 1 (1 α),, Υn 1 (1 α)} T x. Proof Since the lifetime of the k-out-of-(m + n) system is k-max{η 1, η 2,, η m,, τ 1, τ 2,, τ n }, the loss function f(η 1, η 2,, η m, τ 1, τ 2,, τ n ) is T k-max{η 1, η 2,, η m, τ 1, τ 2,, τ n }. It follows from Theorem 1 that the chance distribution of f is Φ(x) Ch{T k-max{η 1, η 2,, η m, τ 1, τ 2,, τ n } x} Ch{k-max{η 1, η 2,, η m, τ 1, τ 2,, τ n } T x} g(x; y 1,, y m )dψ 1 (y 1 ) dψ m (y m ) R m k-max {y 1, y 2,, y m, 1 (1 α), 2 (1 α),, n (1 α)} T x. The theorem is thus proved. 8 Structural System sup { x R m g(x; y 1,, y m )dψ 1 (y 1 ) dψ m (y m ) 1 β} dβ. Theorem 9 Considering a structural system that has n rods in series and an object. Suppose the strength variables of the n rods are independent uncertain variables τ 1, τ 2,, τ n with continuous uncertainty distributions Υ 1, Υ 2,, Υ n, respectively, and the gravity of the object is a random variable η with probability distribution Ψ. If the structural system fails whenever the load variable η exceeds at least one of the strength variables τ 1, τ 2,, τ n, then sup { x + Υ 1 (y x) Υ n (y x)dψ(y) β} dβ.

7 Tail Value-at-Risk in Uncertain Random Environment 7 Proof Since the strength of the structural system is τ 1 τ 2 τ n, the loss function f(η, τ 1, τ 2,, τ n ) is η τ 1 τ 2 τ n. And the chance distribution of f is Φ(x) Ch{η τ 1 τ 2 τ n x} Pr{ω Ω M {η(ω) τ 1 τ 2 τ n x} r}dr M{τ 1 τ 2 τ n y x}dψ(y) (1 Υ 1 (y x) Υ n (y x))dψ(y) + sup Υ 1 (y x) Υ n (y x)dψ(y). { x 1 + Υ 1 (y x) Υ n (y x)dψ(y) 1 β} dβ { + sup x Υ 1 (y x) The theorem is thus verified. Υ n (y x)dψ(y) β} dβ. Theorem 1 Considering a structural system that has n rods in series and an object. Suppose the strength variables of the n rods are independent random variables η 1, η 2,, η n with probability distributions Ψ 1, Ψ 2,, Ψ n, respectively, and the gravity of the object is an uncertain variable τ with continuous uncertainty distribution Υ. If the structural system fails whenever the load variable τ exceeds at least one of the strength variables η 1, η 2,, η n, then sup { x Υ (y 1 y 2 y n + x) R n dψ 1 (y 1 ) dψ n (y n ) 1 β} dβ. Proof Since the strength of the structural system is η 1 η 2 η n, the loss function f(η 1, η 2,, η n, τ) is τ η 1 η 2 η n. And the chance distribution of f is Φ(x) Ch{τ η 1 η 2 η n x} 1 Pr {ω Ω M{τ η 1 (ω) η 2 (ω) η n (ω) x} r}dr M{τ y 1 y 2 y n x} R n dψ 1 (y 1 ) dψ n (y n ) M{τ y 1 y 2 y n + x} R n dψ 1 (y 1 ) dψ n (y n ) Υ (y 1 y 2 y n + x)dψ 1 (y 1 ) dψ n (y n ). R n sup The theorem is thus verified. 9 Conclusion { x Υ (y 1 y 2 y n + x) R n dψ 1 (y 1 ) dψ n (y n ) 1 β} dβ. A tool of value-at-risk was extended to a tool of tail value-at-risk in this paper. The concept of tail value-atrisk is valuable for the uncertain random risk analysis, and some results were showed to illustrate the evaluation of the tail value-at-risk. Furthermore, in order to document the applications for the tail value-at-risk in uncertain random environment, this paper also used the method of tail value-at-risk in parallel system, series system, standby system, k-out-of-(m + n) system and structural system. Compliance with Ethical Standards Funding: This work was supported by National Natural Science Foundation of China under Grant No Conflict of interest: The authors declare that there is no conflict of interest regarding the publication of this paper. Ethical Approval: This article does not contain any studies with human participants performed by any of the authors.

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