The Quasi-Sure Limit of Convex Combinations of Nonnegative Measurable Functions

Abstract

This study shows that, for a sequence of nonnegative valued measurable functions, a sequence of convex combinations converges to a nonnegative function in the quasi-sure sense. This can be used to prove some existence results in multiprobabilities models, and an example application in finance is discussed herein.

1. Introduction

Stochastic models are widely used in practical applications and most are built on a probability space. However, in practice, situations with many possible probabilities occur. A few examples include the uncertain drift or uncertain volatility cases in economic models. These multiprobabilities are called ambiguity, model uncertainty, or Knight uncertainty. Reference [1] emphasized the significant distinction between risk and ambiguity and [2] showed that the distinction between risk and ambiguity is behaviorally significant. To study economic problems while considering ambiguity, mathematical models must be built on a multiprobability space.

Reference [3] first developed the theory of nonlinear -expectation which nontrivially generalizes the classical linear expectation from a probability space to a space with a set of uniformly absolute continuous probabilities. Considering this theory, [4] studied the stochastic differential recursive utility with drift ambiguity. However, many economic and financial problems involve significant volatility uncertainty, which is characterized by a family of nondominated probability measures. Motivated by volatility uncertainty in statistics, risk measures, and super-hedging in finance, [5] introduced a nonlinear expectation, called the G-expectation, which can be regarded as the upper expectation of a specific family of nondominated probability measures. Subsequently, [6, 7] introduced a new type of “G-Brownian motion” and presented the related calculus of Itô’s type. Reference [8] developed a representation of the G-expectation and G-Brownian motion. Reference [9] studied the martingale representation theorem for the G-expectation. Reference [10] determined the properties of hitting times for the G-martingales. References [11, 12] studied the G-BSDEs in G-expectation space.

This paper provides a convergence result in the multiprobability space. In such spaces, under the upper expectation of defined in (4), the corresponding Fatou’s lemma and dominated convergence theorem no longer hold. Therefore, the convergence result reported in this paper will be useful for future studies.

This paper is organized as follows. In Section 2, we prove that, for a sequence of nonnegative measurable functions, there is a sequence of convex combinations which converges to a nonnegative function in the quasi-sure sense. In Section 3, we use the results of Section 2 to prove an existence result in a multiprobabilities model.

2. Main Results

Let be a complete separable metric space, the Borel -algebra of , and the collection of all probability measures on . Let be the space of all -measurable real functions.

Consider a given subset .

Denote

Then, is a Choquet capacity; see [13–15]. A set is called polar if , and we say a property holds “quasi-surely”(q.s.) if it holds outside a polar set. Let , . The sequence is said to converge in capacity to , denoted by , , if The sequence is said to mutually converge in capacity , if

The upper expectation of is defined as follows (see [16] ): for each such that exists for each ,

Lemma 1. Let , . Then,
(i) , q.s., if (ii) is Cauchy q.s., if

Proof. (i) We choose , such that and , . Thus, we obtain q.s.
(ii) Similar to proof (i), we choose , such that and , . Therefore, is Cauchy q.s.

Lemma 2. Let , , be a positive number sequence, and .
(i) If then q.s.
(ii) If then is Cauchy q.s.

Proof. (i) For any , there exists such that, for all , . Then, we have Letting on the right side of the above inequality yields and by Lemma 1, we have q.s.
(ii) Similar to proof (i).

Lemma 3. Let , .
(i) If , then there exists a subsequence of such that q.s.
(ii) If , mutually converges in capacity , then there exist a subsequence of and such that q.s.

Proof. (i) Since , for each , , there exists , such that We can choose . Then, we have By Lemma 2, , q.s.
(ii) Similar to proof (i), there exists a subsequence , which is Cauchy q.s. Then, there exists such that q.s.
If is the space of all -measurable functions in , the sequence mutually converges in capacity , if The results of the above lemmas still hold.

Let denote the convex combination of . Using a similar argument as in Lemma A1.1 of [17], we obtain the following.

Theorem 4. Let be a sequence of valued measurable functions. There exists a sequence such that converges to a valued function q.s.

Proof. Let be defined as . Define asand choose so that where .
It is clear that is a bounded increasing sequence, so there exists such that .
On the compact (metrisable) space , is Cauchy if and only if for each there is so that for all we have or . From the properties of , we have that for there is so that and imply .
For a given , we can take as above and, with the convexity of , we obtain Then, Without loss of generality, we can set . By (16), . Taking the expectation about each and letting , we obtain Thus, we have Therefore, That is, mutually converges in capacity , and, by Lemma 3, there exist a subsequence and a valued function such that , q.s.

3. Application of Theorem 4

We set, for , and denote .

Consider the following optimization problem in finance:where is the terminal wealth of the hedging portfolio at terminal time , , is the nonnegative contingent claim which the investor attempts to hedge, , is the loss function which is an increasing convex function defined on , and is the constraint regarding the initial wealth. The corresponding hedging problem in the single probability model was introduced in [18] and is referred to as efficient hedging.

We use the result of the Theorem 4 to prove the existence of the solution of problem (25).

Theorem 5. There is a solution to problem (25).

Proof. Let consist of elements of that satisfy and let be a minimizing sequence for (25) in . By Theorem 4, there exists a sequence belonging to such that , q.s. Since , i.e., and , we obtain , , and where the second equality sign is a result of the dominated convergence theorem under probability . Therefore, .
Similarly, since , q.s., we have By the convexity of and function and belonging to , we can conclude that is not larger than the corresponding convex combination of , . And because is a minimizing sequence for (25) in , Then we have So

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 11301011 and the Beijing Natural Science Foundation under Grant no. 1112009.