Sharp Large Deviation for the Energy of -Brownian Bridge
We study the sharp large deviation for the energy of -Brownian bridge. The full expansion of the tail probability for energy is obtained by the change of measure.
We consider the following -Brownian bridge: where is a standard Brownian motion, , , and the constant . Let denote the probability distribution of the solution of (1). The -Brownian bridge is first used to study the arbitrage profit associated with a given future contract in the absence of transaction costs by Brennan and Schwartz .
-Brownian bridge is a time inhomogeneous diffusion process which has been studied by Barczy and Pap [2, 3], Jiang and Zhao , and Zhao and Liu . They studied the central limit theorem and the large deviations for parameter estimators and hypothesis testing problem of -Brownian bridge. While the large deviation is not so helpful in some statistics problems since it only gives a logarithmic equivalent for the deviation probability, Bahadur and Ranga Rao  overcame this difficulty by the sharp large deviation principle for the empirical mean. Recently, the sharp large deviation principle is widely used in the study of Gaussian quadratic forms, Ornstein-Uhlenbeck model, and fractional Ornstein-Uhlenbeck (cf. Bercu and Rouault , Bercu et al. , and Bercu et al. [9, 10]).
In this paper we consider the sharp large deviation principle (SLDP) of energy , where
Our main results are the following.
Theorem 1. Let be the process given by the stochastic differential equation (1). Then satisfies the large deviation principle with speed and good rate function defined by the following: where .
Theorem 2. satisfies SLDP; that is, for any , there exists a sequence such that, for any , when approaches enough, where The coefficients may be explicitly computed as functions of the derivatives of and (defined in Lemma 3) at point . For example, is given by with , and .
2. Large Deviation for Energy
Given , we first consider the following logarithmic moment generating function of ; that is, And let be the effective domain of. By the same method as in Zhao and Liu , we have the following lemma.
Lemma 3. Let be the effective domain of the limit of ; then for all , one has with where and . Furthermore, the remainder satisfies
Proof. By Itô’s formula and Girsanov’s formula (see Jacob and Shiryaev ), for all and , Therefore, If , we can choose such that . Then where , and . Therefore,
Proof of Theorem 1. From Lemma 3, we have and is steep; by the Gärtner-Ellis theorem (Dembo and Zeitouni ), satisfies the large deviation principle with speed and good rate function defined by the following:
3. Sharp Large Deviation for Energy
For , let Then where is the expectation after the change of measure
By Lemma 3, we have the following expression of .
Lemma 5. For all , when approaches enough, For , one gets the following.
Lemma 6. For all , the distribution of under converges to distribution. Furthermore, there exists a sequence such that, for any when approaches enough,
It only remains to prove Lemma 6. Let be the characteristic function of under ; then we have the following.
Lemma 7. Whenapproaches,belongs to and, for all, Moreover, with where for some positive constant, andis some positive constant.
Proof. For any, By the same method as in the proof of Lemma 2.2 in  by Bercu and Rouault, there exist two positive constantsandsuch that therefore, belongs to, and by Parseval’s formula, for some positive constant, let we get whereis some positive constant.
Proof of Lemma 6. By Lemma 3, we have
Noting that, and
for any by Taylor expansion, we obtain
therefore, there exist integers, and a sequenceindependent of ; whenapproaches, we get
whereis uniform as soon as.
Finally, we get the proof of Lemma 6 by Lemma 7 together with standard calculations on thedistribution.
This research was supported by the National Natural Science of Tianyuan Foundation under Grant 11226202.
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