Journal of Function Spaces
Volume 2019 (2019), Article ID 3695869, 4 pages
https://doi.org/10.1155/2019/3695869

Research Article

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

Yulian Fan

School of Science, North China University of Technology, Beijing 100144, China

Correspondence should be addressed to Yulian Fan; fanyl@ncut.edu.cn

Received 24 September 2018; Revised 14 November 2018; Accepted 21 November 2018; Published 1 January 2019

Academic Editor: Pasquale Vetro

Copyright © 2019 Yulian Fan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 as

and 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.

References

F. H. Knight, Risk, Uncertainty and Profit, Dover Publications, Boston, MA, USA, 1921.

D. Ellsberg, “Risk, ambiguity, and the savage axioms,” The Quarterly Journal of Economics, vol. 75, no. 4, pp. 643–669, 1961.

S. Peng, “Backward SDE and related g-expectation,” in Backward stochastic differential equations (Paris, 1995-1996), vol. 364 of Pitman Res. Notes Math. Ser., pp. 141–159, Longman, Harlow, UK, 1997.

Z. Chen and L. Epstein, “Ambiguity, risk, and asset returns in continuous time,” Econometrica, vol. 70, no. 4, pp. 1403–1443, 2002.

S. Peng, “G-Expectation, G-Brownian Motion and related stochastic calculus of Itô’s type,” in Proceedings of the The Abel Symposium 2005, Abel Symposia 2, Benth et al., Ed., pp. 541–567, Springer.

S. Peng, “Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation,” Stochastic Processes and Their Applications, vol. 118, no. 12, pp. 2223–2253, 2008.

S. G. Peng, “Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations,” Science China Mathematics, vol. 52, no. 7, pp. 1391–1411, 2009.

M. Hu and S. Peng, “On the representation theorem of G-expectation and paths of G-Brownian motion,” Acta Mathematicae Applicatae Sinica, vol. 25, no. 3, pp. 539–546, 2009.

H. M. Soner, N. Touzi, and J. Zhang, “Martingale representation theorem for the G-expectation,” Stochastic Processes and Their Applications, vol. 121, no. 2, pp. 265–287, 2011.

Y. Song, “Properties of hitting times for G-martingales and their applications,” Stochastic Processes and Their Applications, vol. 121, no. 8, pp. 1770–1784, 2011.

M. Hu, S. Ji, S. Peng, and Y. Song, “Backward stochastic differential equations driven by G-Brownian motion,” Stochastic Processes and Their Applications, vol. 124, no. 1, pp. 759–784, 2014.

M. Hu, S. Ji, S. Peng, and Y. Song, “Comparison theorem, Feynman–Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion,” Stochastic Processes and Their Applications, vol. 124, no. 2, pp. 1170–1195, 2014.

C. Dellacherie, Capacitis et Processus Stochastiques, Springer, Berlin, Germany, 1972.

G. Choquet, “Theory of capacities,” Annales de l'Institut Fourier, vol. 5, pp. 131–295, 1955.

L. Denis, M. Hu, and S. Peng, “Functional spaces and capacity related to a sulinear expectations: application to G-Brownian motion paths,” Potential Analysis, vol. 34, no. 2, pp. 139–161, 2011.

P. J. Huber and V. Strassen, “Minimax tests and the Neyman-Pearson lemma for capacities,” The Annals of Statistics, vol. 1, pp. 251–263, 1973.

F. Delbaen and W. Schachermayer, “A general version of the fundamental theorem of asset pricing,” Mathematische Annalen, vol. 300, no. 3, pp. 463–520, 1994.

H. Föllmer and P. Leukert, “Efficient hedging: cost versus shortfall risk,” Finance and Stochastics, vol. 4, no. 2, pp. 117–146, 2000.