Abstract
Up to now, most of the academic researches about the large deviation and risk theory are under the framework of the classical linear expectations. But motivated by problems of model uncertainties in statistics, measures of risk, and superhedging in finance, sublinear expectations are extensively studied. In this paper, we obtain a type of large deviation principle under the sublinear expectation. This result is a new expression of the Gärtner-Ellis theorem under the sublinear expectations which is in the classical theory of large deviations. In addition, we introduce a new process under the sublinear expectations, that is, the -Poisson process. We give an application of our result and obtain the rate function of the compound -Poisson process in the upper bound of large deviations for capacities. The application of our result opens a new field for the research of risk theory in the future.
1. Introduction
Large deviation theory is one of the key techniques of modern probability, a role which is emphasized by the recent award of the Abel prize to S.R.S. Varadhan, one of the pioneers of the subject. The large deviation principle characterizes the limiting behavior as of a family of probability measures in terms of a rate function. Also Cramér’s theorem has been widely known for a long time as a fundamental result in large deviations. It is very useful in many fields. But Cramér’s theorem is limited to the i.i.d. case. However, a glance at the proof should be enough to convince the reader that some extension to the non-i.i.d. case is possible. As described in [1], Gärtner-Ellis theorem is a generalization of Cramér’s theorem in non-i.i.d situation to conclusions.
Motivated by problems of model uncertainties in statistics, measures of risk, and superhedging in finance, sublinear expectations are extensively studied [2]. Since the paper [3] on coherent risk measures, authors have been more and more interested in sublinear expectations [4, 5]. By Peng [6], we know that a sublinear expectation can be represented as the upper expectation of a set of linear expectations ; that is, . In most cases, this set is often treated as an uncertain model of probabilities and the notion of sublinear expectation provides a robust way to measure a risk loss . In fact, nonlinear expectation theory provides many rich, flexible, and elegant tools and plays an important role in many aspects. In particular, its important application in stochastic dominance, stochastic differential game, financial mathematics, economics, and partial differential equations attracted a large number of mathematicians, economists, and financial experts to join the research, for instance, the application of nonlinear expectation in the dynamic measurement and dominance of financial risk, backward stochastic differential equation theory and its application in financial products innovation, pricing, and so forth. We can see its recent developments from the following literature [7–12].
In this paper, we are interested in where is a set of probability measures, especially set , . Obviously, is a capacity. Under the sublinear expectation, the upper bound of Cramér’s theorem has come to a conclusion similar to the linear expectation (see [13]). On this basis, additionally, the main aim of this paper is to obtain Gärtner-Ellis’s upper bound for the capacity .
This paper is organized as follows. In Section 2, we give some notions and lemmas that are useful in this paper. In Section 3, we give the main result including the proof. In Section 4, we give a brief application of our result in the classical risk model.
2. Preliminaries
We present some preliminaries in the theory of sublinear expectations. More details of this section can be found in Peng [6, 14, 15].
Definition 1. Let be a given set and let be a linear space of real valued functions defined on . We assume that all constants are in and that implies . is considered as the space of our “random variables.” A nonlinear expectation on is a functional satisfying the following properties: for all , one has(a)monotonicity: if , then ;(b)constant preserving: , .
The triple is called a nonlinear expectation space (compare with a probability space ). We are mainly concerned with sublinear expectation where the expectation satisfies also(c)subadditivity: ;(d)positive homogeneity: , .
If only (c) and (d) are satisfied, is called a sublinear functional.
The following representation theorem for sublinear expectations is very useful (see Peng [6, 15] for the proof).
Lemma 2. Let be a sublinear functional defined on ; that is, (c) and (d) hold for . Then there exists a family of linear functionals on such that
If (a) and (b) also hold, then are linear expectations for . If we make, furthermore, the following assumption. (H1)For each sequence such that for , one has .
Then for each , there exists a unique (-additive) probability measure defined on such that
In this paper, we are interested in the following sublinear expectation: where is a set of probability measures. Let be a given set and let be a -algebra. Define , ; then is a capacity.
Let denote the space of continuous functions defined on .
Now we recall some important notions of sublinear expectations distributions (see Peng [6, 14, 15]).
Definition 3. Let and be two random variables in a sublinear expectation space . They are called identically distributed, denoted by , if for , and exist; then one has
Definition 4. In a sublinear expectation space , a random vector is said to be independent of another random vector , if for , and exist; then one has
We conclude this section with some notations on large deviations under a sublinear expectation [16].
Let be a topology space and be a -algebra on . Let be a family of measurable maps from into and , be a positive function satisfying as . A nonnegative function on is called a (good) rate function if is (compact) closed for all .
is said to satisfy large deviation principle (LDP) with speed and with rate function if for any measurable closed subset , and for any measurable open subset , Equations (7) and (8) are referred, respectively, to as upper bound of large deviations (ULD) and lower bound of large deviations (LLD).
is said to be exponentially tight if for any , there exists a compact set such that
is said to satisfy -upper bound of large deviations with speed and with rate function if (7) for any compact subset .
It is known that if satisfies -large deviation principle with speed and with rate function and is exponentially tight, then it satisfies large deviation principle with speed and with rate function .
Definition 5. For any rate function and any , the -rate function is defined as While in general is not a rate function, its usefulness stems from the fact that for any set , Consequently, the upper bound in (7) is equivalent to the statement that for any and for any measurable set ,
3. Main Result
In this section, firstly let us present some notations and assumptions that are used in the following details.
Consider a sequence of random vectors ; let be identically distributed under , where possesses logarithmic moment generating function , . We also assume that each is independent of for under . Denote .
Specifically, the following assumption is imposed throughout this section.
Assumption 6. For each , the logarithmic moment generating function, defined as the limit exists as an extended real number. Furthermore, the origin belongs to the interior of .
Define where is the Fenchel-Legendre transform of , with .
We always assume that(H2)if , then .
Lemma 7. Let be a fixed integer. Then, for every ,
Proof. First note that for all , Since is fixed, as and
The next lemma describes a property of , which will be used to give a more accurate expression of the rate function.
Lemma 8. If for some , then for any , and for any , .
Here, we omit the proof of Lemma 8 (refer to [13] or [17]).
The following theorem is the main result of this paper.
Theorem 9. Let Assumption 6 hold. Then we have for any closed set , where is a convex rate function.
Proof. As mentioned in Section 2, establishing the upper bound is equivalent to proving that for every and every closed set ,
where is the -rate function associated with . Fix a compact set . For every , choose such that
This is feasible on account of the definitions of and . For each , choose such that and let be the ball with center at and radius . Observe for every , , and measurable that
In particular, for each and ,
Also, for any ,
and therefore,
Since is compact, one may extract from the open covering of a finite covering that consists of such balls with centers in . By the union of events bound and the preceding inequality,
Hence, by our choice of ,
Since , the upper bound (7) is established for all compact sets.
As described earlier in Section 2, the upper bound of large deviations is extended to all closed subsets of by showing that is an exponentially tight family of probability measures. Let . Since , the union of events bound yields
where are the coordinates of the random vector ; namely, are the laws governing . Let denote the th unit vector in for . Since (refer to [13, Lemma 3.1]), there exist and such that and for . By Chebyshev’s inequality, we have
Hence, for ,
Consequently, by the union of events bound and Lemma 7,
Therefore, is an exponentially tight family of probability measures, since the hypercubes are compact.
Remark 10. Since is not linear, Cramér’s method is not useful for lower bound of large deviations. This is consistent with the conclusion of [13]. In the paper [13], the author gives a counter example to illustrate that under the sublinear expectation, the lower bound of Cramér’s theorem is not obtained. Since Gärtner-Ellis theorem be a generalization of Cramér’s theorem in non-i.i.d situation to conclusions, we see under the assumptions of theorem, the lower bound of Gärtner-Ellis theorem does not hold.
4. Application
In this section, we consider the classical risk process under sublinear expectation . The classical risk process is defined by where is the initial capital and is the (constant) premium rate, and the aggregate claims process is a compound Poisson process. More precisely we have and , where is a sequence of positive random variables, is a counting process with points , , and independent, the , , are independent and identically distributed, and where is a Poisson process with intensity under linear expectation. Now we consider is a -Poisson process (its definition refers to [18]) under sublinear expectation . Then, is a compound -Poisson process correspondingly.
We also assume the following superexponential condition holds for the random variables under sublinear expectation .
Assumption 11. for all .
Let be the moment generating function (m.g.f.) of ; that is, Then its logarithmic moment generating function is expressed as follows:
Let be the limit of the normalized logarithmic moment generating function of ; that is,
In order to obtain the m.g.f. of the process , firstly we introduce a lemma which plays a role in the next lemma. We omit its proof which can be found in [19, Lemma 1.1].
Lemma 12. If a sequence of -dimensions random variables under sublinear expectation space satisfies for any , is independent of , then the following conclusions are established.(1)If are lower semicontinuous functions in , one has (2)If are upper semicontinuous functions in and there exists a continuous function such that is quasicontinuous and , and , for any , one has
Lemma 13. If and are independent, then for each
Proof. By Lemma 12, we have This completes the proof.
Then we can see That is to say, the normalized logarithmic moment generating function of has a limit. By Theorem 9 that we obtained in Section 3, we can say satisfies the upper bound of Gärtner-Ellis theorem with rate function defined by
Next, we give you a brief description about -Poisson process in Ren’s Ph.D. thesis [17]. Let be -Poisson process under sublinear expectation . Then, satisfies the following one-dimensional equation: where , . Referring to [18], we know, for any increasing function , and for any decreasing function , Then we have , for any , and , for any .
Since represents the amount claimed, we know and ; and so . In the above formula (42), we set ,
Substituting this result into (39), we have By (40) and Lemma 8, we get the rate function of in the upper bound of Gärtner-Ellis theorem. As described below, for , The above result can be calculated further, specifically according to the different distributions of the amount claimed.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.