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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-at-risk eventually lies between the upper and lower value-at-risks quasi surely.

In financial industry, the value-at-risk has been one of main tools for risk management (see, e.g., McNeil et al. [

To capture the distribution uncertainty, the theory of sublinear expectation is introduced and developed (see Peng [

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 value- at-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.

In this section, we recall the basis of the sublinear expectation, introduced by Peng [

for some

We consider a sublinear expectation

1) Monotonicity: if

2) Constant preserving:

3) Subadditivity:

4) Positive homogeneity:

Moreover, we assume that

where

define capacities, where 1_{A} denotes the indicator function of a set A. That is, each

1)

2) If

We refer to Denenberg [

3) If

4) If

Let us recall several concepts in the sublinear expectation theory. The random variable

We say that

A sequence

Then if

For any

Proposition 3.1 Let

1)

2)

3)

Proof. The assertion (1) follows from the monotonicity of

To prove (2), take

Similarly,

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

For an IID sequence

If

Indeed,

We show a stronger result, which is a generalization of Glivenko-Cantelli lemma.

Theorem 3.3 Let

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

where

Proof. Let

By this recursion, we can find

With the help of Lemma 3.4, we can show Theorem 3.3.

Proof of Theorem 3.3. Let

By (2), we have,

Thus,

where for

So we have

Now, for any

so

Therefore, on

If we write

meaning the assertion of the theorem. □

Recall that for a function

Then we have the following:

Theorem 3.5 Let

Suppose that for

Then, the historical value-at-risk eventually lies in between the upper and lower value-at-risk, i.e.,

Proof. Consider the event A defined by

In view of Theorem 3.3, it suffices to show that for a given

To this end, fix

Thus we can take

Next, take

and set

So we have

leading to