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Weak and Strong Limit Theorems for Stochastic Processes under Nonadditive Probability
This paper extends laws of large numbers under upper probability to sequences of stochastic processes generated by linear interpolation. This extension characterizes the relation between sequences of stochastic processes and subsets of continuous function space in the framework of upper probability. Limit results for sequences of functional random variables and some useful inequalities are also obtained as applications.
Laws of large numbers are the cornerstones of theory of probability and statistics. As we know, under appropriate assumptions, the well-known strong law of large numbers (SLLN for short) states that for a sequence of random variables , its sample mean converges to a unique constant almost surely in the framework of probability. But many empirical analyses and theoretical works show us that nonadditive probability and nonlinear expectation are very probably faced in economics, finance, number theory, statistics, and many other fields, such as capacity, Choquet integral (see Choquet ), (nontrivial) -probability, (nontrivial) -expectation (see El Karoui et al. ), and -expectation (see Peng ). And for each nonadditive probability, say , we can define many different expectations related to , denoted by . For nonlinear , random variables may have mean uncertainty; that is, , or variance uncertainty; that is, . In such cases, there are many scholars that investigate the limit theorems under or , such as the laws of large numbers, laws of iterated logarithm, central limit theorems under either or , and other related problems. One can refer to Peng [4, 5], Chen and Hu , Wu and Chen , the papers mentioned in the following, and some references therein.
When has mean uncertainty, sample mean probably cannot converge to a unique constant almost everywhere (shortly a.e., which should be well defined) under a nonadditive probability or a set of probabilities. Marinacci , Teran , and some of the references therein investigate the SLLN via Choquet integrals related to completely monotone capacity . They suppose that is a sequence of independent and identically distributed random variables under capacity and prove that all the limit points of convergent subsequences of sample mean belong to an interval with probability 1 (w.p. 1 for short) under ; that is,
Recently, Chen  and Chen et al.  prove the SLLN via a sublinear expectation . They suppose that is a sequence of independent random variables under (see Peng ) and prove that where is the lower probability (see Halpern ) corresponding to , and . It is obvious that is a subset of . On the other hand, Chen  and Chen  prove that any element of is the limit of certain convergent subsequence of sample mean w.p. 1 under upper probability corresponding to .
This paper is motivated by the problem of limit theorems of sequences of stochastic processes in the framework of nonadditive probabilities and the estimation of expectations of functionals of stock prices with ambiguity. If there is no mean uncertainty, they are trivial. But if there is mean uncertainty, then as the SLLN of random variables under nonadditive probability behaves, limit theorems related to stochastic processes become interesting and different from classical case. Chen  investigates a limit theorem for -quadratic variational process in the framework of -expectation. More generally, for random variables with mean uncertainty, in the framework of upper and lower probabilities , we consider a simple sequence of stochastic processes generated by linearly interpolating at . X. Chen and Z. Chen  prove that all the limit points of subsequences of are elements of w.p. 1 under lower probability ; namely, where is a subset of continuous function space on (see Section 2). Conversely, for any element of , is it a limit point of certain subsequence of w.p. 1 under upper probability ? In other words, does the following statement hold true?
In this paper we will employ the independence condition of Peng  to investigate this problem and prove that under certain conditions it holds true. We will see that this strong form can be implied by a weak form (see Section 4). Under continuous upper probability our strong limit theorem becomes weaker than our weak one. From the face of this meaning it is different from classical framework. But in fact, it coincides with the classical case. We also extend our strong limit theorem to functional random variables and show some useful inequalities under continuous upper probability .
The remaining part of this paper is organized as follows. In Section 2 we recall some basic definitions and properties of lower and upper probabilities. And we will also give basic assumptions for all of the subsequent sections. Some auxiliary lemmas are proved in Section 3. In Section 4, we prove a weak limit theorem under general upper probability. Section 5 mainly presents a strong limit theorem under continuous upper probability and its extension to functional random variables. In Section 6 we give a simple example as applications in finance.
2. Basic Settings
Let be a nonempty set. denotes a -algebra of subsets of . Let be a pair of nonadditive probabilities, related to a set of probabilities on measurable space , given by
It is obvious that upper probability and lower probability are conjugate capacities (see Choquet ); that is, normalization: , ; monotonicity: for all , if , then and ; conjugation: for all , , where denotes the complementary set of .
Moreover, we can easily get the following properties which are useful in this paper (see also Chen et al. ).
Proposition 1. For any sequence of sets , , we have the following.(i)Subadditivity of : .(ii)Lower continuity of : if , then .(iii)Upper continuity of : if , then .(iv)If for all , then .
We say upper probability (resp., lower probability ) is continuous if and only if it is upper and lower continuous. Obviously, upper probability is continuous if and only if lower probability is continuous.
The corresponding pair of upper and lower expectations of is given as follows: where denotes the set of all real-valued random variables on such that . Obviously, is a sublinear expectation (see Peng ).
Definition 2 (see Peng ). Let be a sequence of random variables on in . We say it is a sequence of independent random variables under upper expectation , if for all real-valued continuous functions on , denoted by , with linear growth condition; that is, there exists a constant s.t. we have
Throughout this paper we assume (unless otherwise specified) that is a sequence of independent random variables under upper expectation satisfying for all , respectively, where .
Set and for any . We define a sequence of stochastic processes by linearly interpolating at for each and ; that is, where denotes the greatest integer which is less or equal to a nonnegative number .
Let be a linear space of all real-valued continuous functions on with supremum as its norm, denoted by . Let be a subset of such that all the functions are absolutely continuous on with and almost everywhere on . Thus, we can easily have the following.
Proposition 3. is compact.
3. Auxiliary Lemmas
Before investigating the convergence problem of sequence under upper probability, in this section we first give some useful lemmas.
Definition 4. A set is said to be a polar set if . We say an event holds quasisurely (q.s. for short) if it holds outside a polar set.
We first give the following property.
Lemma 5. The sequence of functions on is relatively compact w.p. 1 under lower probability .
Proof. For each , function can be rewritten as
Obviously, for each , , and for any , the first-order derivative of with respect to for every is
Then the difference of with respect to follows that for any with ,
From , we have . Thus, we can get an upper bound of the norm of as follows:
In addition, for any such that , we can get from (13) that
In fact, without loss of generality, we assume that , if ; thus, ; then from (13) it follows that
Otherwise if , thus , which implies that ; then from (13) we have
Hence, from (16) and (17) we know that (15) holds true. Thus, we can easily get that is equicontinuous with respect to w.p. 1 under lower probability from property (iv) of Proposition 1. Together with (14) this sequence is relatively compact in w.p. 1 under . We get the desired result.
The following lemma is very useful in the proofs of our main theorems and its proof is similar as Theorem 3.1 of Hu . Here we omit its proof.
Lemma 6. Given a sequence of independent random variables under , we assume that there exist two constants such that and for all , and we also assume that . Then for any increasing subsequence of satisfying converges to as tends to , and for any with linear growth, we have where for all .
4. Weak Limit Theorem
In this section we will investigate the weak convergence problem of under general upper probability.
Theorem 7. For any and , there exists a subsequence such that where is an increasing subsequence of and depends on , , and .
Proof. For any and , by Lemma 5 we only need to find a subsequence satisfying (19). Set
Note that for any integer ,
Denoting , since , thus, for all , , for all . Hence, taking we have
Let be a positive integer for any ; then by the definition of (see (10)), it follows that for , and ,
In addition, since , we know that
Then it follows that
For and , we set
Obviously, is a continuous function on satisfying linear growth condition. Since is independent under (see Definition 2), from (25), we have
Since, for all with and , , and , let tend to as tends to ; then by Lemma 6 we have since for . Thus from (27) and (28) it follows that . Obviously, for all . Hence this theorem follows.
Corollary 8. Let be a real-valued continuous functional on ; then for any and , there exists a subsequence such that
where is an increasing subsequence of and depends on , , and .
In particular, if we assume that for all , then we have where .
5. Strong Limit Theorem under Continuous Upper Probability
In the previous Sections 2–4, we consider the general upper probability . For the sake of technique, in this section we further assume that is continuous and investigate a strong limit theorem of under such a continuous upper probability and its extension.
5.1. Strong Limit Theorem
Theorem 9. Any is a limit point of some subsequence of w.p. 1 under ; that is, where denotes the cluster set of all the limit points of real sequence .
Proof. From Lemma 5, since is continuous, we only need to prove that for any and any ,
Let and be defined the same as in the proof of Theorem 7. Then it is sufficient to prove that for any fixed we can find a subsequence of such that
Take for , where is an integer. From Theorem 7 and the continuity of we can get
Thus this theorem is proved.
Remark 10. From the proof of Theorem 9 we can see that it is implied by weak limit Theorem 7 under continuous upper probability. It seems that “weak limit theorem” is stronger than “strong limit theorem” under continuous upper probability. If is a singleton, thus we have . Then our “strong limit theorem” is not the same form as the strong law of large numbers for sequences of random variables, since the former form is related to inferior limit and the latter one is related to limit.
5.2. Extension to Functional Random Variables
By Theorem 9 we can easily get the following limit result for functional random variables.
Corollary 11. Let be a real-valued continuous functional defined on ; then we have, for any ,
From the proof of Theorem 3.1 and Corollary 3.2 of Chen et al.  the following lemma can be easily obtained.
Lemma 12. Supposing is a real-valued continuous function on , then
Corollary 13. Let be defined the same as Lemma 12, then
Especially, if we assume , for all , then
Proof. Take , . It is easy to check that is a continuous functional on , and obviously . For any , . Thus, from Corollary 11 it follows that
Then this corollary follows from (37) of Lemma 12 and (40).
In this subsection we will give some useful examples as applications in inequalities.
Example 14. Let be a Lebesgue integrable function defined from to ; we denote . Then
hold w.p. 1 under , respectively, where
Especially for , we have w.p. 1 under , respectively,
Proof. Observe that for all is a continuous functional defined from to . And it is easy to check that w.p. 1 under ,
By Corollary 11 we know that w.p. 1 under
Since for any , almost everywhere for , then note that, for all ,
Thus, inequality (41) holds w.p. 1 under . The proof of inequality (42) is similar to inequality (41) and inequalities (44) are obvious. We complete the whole proof.
Example 15. For any integer , we have that hold w.p. 1 under , respectively.
Proof. It is easy to check that is a continuous functional on . Thus, this example can be similarly proved as Example 14.
6. Applications in Finance
We consider a capital market with ambiguity which is characterized by a set of probabilities, denoted the same as previous sections by such that the corresponding upper probability is continuous. For simplicity, let risk free rate be zero. We will investigate the stock price over time interval on the measurable space , and we assume that the increments of stock price in time period is independent from for all ; that is, for each probability , and are mutually independent under for all . We also assume that the price of the stock is uniformly bounded with respect to and the largest and smallest expected average return of this stock over time interval are and , respectively; that is, where , and .
For any , take , and let and for . Then it is obvious that is a sequence of independent random variables in under upper probability , with supermean and submean for all . Denote the average stock price of by ; then Then by inequalities (41) and (42) it follows that hold, respectively, w.p. 1 under continuous upper probability . (Together with X. Chen and Z. Chen  we will see that these two inequalities can become equalities in the future.)
7. Concluding Remarks
This paper proves that any element of subset of continuous function space on is a limit point of certain subsequence of stochastic processes in upper probability and with probability 1 under continuous upper probability. It is an extension of strong law of large numbers from random variables to stochastic processes in the framework of upper probability. The limit theorem for functional random variables also is proved. It is very useful in finance when there is ambiguity. But the constraint conditions in this paper are very strong, such as the condition and independence under sublinear expectation. How can we weaken the constraint conditions? Does the strong limit theorem under upper probability still hold without continuity of We will investigate them in the future work.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was partially supported by the WCU (World Class University) Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (R31-20007) and was also partially supported by the National Natural Science Foundation of China (no. 11231005).
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