On Historical Value at Risk under Distribution Uncertainty

We investigate the asymptotics of the historical value-at-risk under capacities defined by sublinear expectations. By generalizing Glivenko-Cantelli lemma, we show that the historical value-atrisk eventually lies between the upper and lower value-at-risks quasi surely.


Introduction
In financial industry, the value-at-risk has been one of main tools for risk management (see, e.g., McNeil et al. [1], and Föllmer and Schied [2]).In this framework, the random variables for assets or asset returns are assumed to have distributions without uncertainty.In other words, it is implicitly assumed that there are true asset distributions and the estimation difficulty comes from our limited capability.However, it should be remarked that there is a possibility that the assets have the distribution uncertainty, i.e., the assets may have Knightian uncertainty (see Knight [3]).
To capture the distribution uncertainty, the theory of sublinear expectation is introduced and developed (see Peng [4] [5] and the references therein).In this theory, the term probability is replaced by the ones of the upper and lower capacities induced by the upper and lower expectations, respectively, and the distribution uncertainty is described by the gap between the upper and lower expectations.
In this paper, we consider the value-at-risk type risk measure under the sublinear expectation, where the reference probability measure in the classical framework is replaced by the upper and lower capacities.We call these the upper and lower value-at-risk, respectively.Our aim is to study the asymptotic behavior of the historical value-at-risk under uncertainty.In doing so, we prove a generalization of Glivenko-Cantelli lemma under uncertainty, and then show that the historical value-at-risk eventually lies in between the upper and lower valueat-risks quasi surely.
This paper is organized as follows: In Section 2, we recall the theory of sublinear expectation.Section 3 is devoted to the statement of the main results and its proofs.

Sublinear Expectation and Capacities
In this section, we recall the basis of the sublinear expectation, introduced by Peng [4].Let Ω be a given set and  a linear space of  -valued functions on Ω.We assume that ( ) and ϕ is a bounded function on n  or ( ) for some 0 C > and m ∈  depending on ϕ .We call an element in  a random variable.
We consider a sublinear expectation : , in the sense of [4].Namely, E is assumed to be satisfy the following conditions: for any , 2) Constant preserving: [ ] Moreover, we assume that [ ] Then, by Theo- rem 2.1 and Remark 2.2 in [5], there exists a set  of probability measures on ( ) ( ) where P E denotes the linear expectation with respect to P ∈  .Then, the [ ] define capacities, where 1 A denotes the indicator function of a set A. That is, each satisfies the following: We refer to Denenberg [6] for the theory of capacities.Throughout this paper, we assume that each satisfies the following: 3) If { } ( ) Let us recall several concepts in the sublinear expectation theory.The random variable n Y is said to be independent of ( ) is called the one of independent, identically distributed random variables if i X and 1 X have the same distribution, and if for any 1 i ≥ .As in the linear case, we call a sequence of independent, identically distributed random variables an IID sequence.We say that the distribution of X has an uncertainty if ( ) Then if µ µ ≠ , then we say that X has the mean uncertainty.Similarly, X is said to have the volatility or variance uncertainty if σ σ ≠ .
To prove (2), take , , . It follows from the definition of the capacity that ( ) ( ) x F x →∞ = follows.Finally, by an argument similar to the proof of (2) with ( ] . □ The proposition above justifies the following definition: Definition 3.2 For a random variable X, we call the function F * and F * as in (1) the upper and lower cumulative distribution functions of X, respectively.
for some 1 p > , then by Strong law of large number under sublinear ex- pectation (see Theorem 1 in Chen [7]), Indeed, We show a stronger result, which is a generalization of Glivenko-Cantelli lemma.
for some 1 p > .Denote by F * and F * the upper and lower cumulative distribution functions of X respectively, and denote by ( ) We need the following lemma for the proof of the theorem.Lemma 3.4 Under the assumtions imposed in Theorem 3.3, for ( ) , , 0 1, where ( ) ( ) Proof.Let ( ) ε > be fixed.By Lemma 3.4, there exists a partion of  such that , , 0 1.
By ( 2), we have, Thus, ( ) 1 q P A = for any P ∈  and so ( ) for any P ∈  .Hence


Now, for any x ∈  there exists j such that . Thus, by , ˆˆˆ.
If we write B ε for the event inside the brace above and denote Then we have the following: for some 1 p > .Denote by F * and F * the upper and lower cumulative distribution functions of X respectively, and denote by ( ) Consider the upper, lower, and historical value-at-risk defined respectively by , VaR : , VaR : .
Suppose that for ( ) Then, the historical value-at-risk eventually lies in the upper and lower value-at-risk, i.e., In view of Theorem 3.3, it suffices to show that for a given 0 ε > and A ω ∈ there exists ( ) To this end, fix A ω ∈ and set ( ) ( ) . By the definition of the infimum and the condition (4), ( ) ( ) and set k t = +∞ .With this sequence, the lemma follows.□ With the help of Lemma 3.4, we can show Theorem 3.3.Proof of Theorem 3.3.Let 0