Sharp Large Deviation for the Energy of 𝜶-Brownian Bridge

Abstract

We study the sharp large deviation for the energy of 𝛼-Brownian bridge. The full expansion of the tail probability for energy is obtained by the change of measure.

1. Introduction

We consider the following 𝛼-Brownian bridge: 𝑑𝑋_𝑡𝛼=−𝑋𝑇−𝑡^𝑡𝑑𝑡+𝑑𝑊_𝑡,𝑋₀=0, (1) where 𝑊 is a standard Brownian motion, 𝑡∈[0,𝑇), 𝑇∈(0,∞), and the constant 𝛼>1/2. Let 𝑃_𝛼 denote the probability distribution of the solution {𝑋_𝑡,𝑡∈[0,𝑇)} of (1). The 𝛼-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 [1].

𝛼-Brownian bridge is a time inhomogeneous diffusion process which has been studied by Barczy and Pap [2, 3], Jiang and Zhao [4], and Zhao and Liu [5]. They studied the central limit theorem and the large deviations for parameter estimators and hypothesis testing problem of 𝛼-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 [6] 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 [7], Bercu et al. [8], and Bercu et al. [9, 10]).

In this paper we consider the sharp large deviation principle (SLDP) of energy 𝑆_𝑡, where 𝑆_𝑡=^𝑡₀𝑋²_𝑠(𝑠−𝑇)²𝑑𝑠. (2)

Our main results are the following.

Theorem 1. Let {𝑋_𝑡,𝑡∈[0,𝑇)} be the process given by the stochastic differential equation (1). Then {𝑆_𝑡/𝜆_𝑡,𝑡∈[0,𝑇)} satisfies the large deviation principle with speed 𝜆_𝑡 and good rate function 𝐼(⋅) defined by the following: 𝐼(𝑥)=18𝑥2𝛼₀−1𝑥−1²,𝑖𝑓𝑥>0;+∞,𝑖𝑓𝑥≤0, (3) where 𝜆_𝑡=log(𝑇/(𝑇−𝑡)).

Theorem 2. {𝑆_𝑡/𝜆_𝑡,𝑡∈[0,𝑇)} satisfies SLDP; that is, for any 𝑐>1/(2𝛼−1), there exists a sequence 𝑏_𝑐,𝑘 such that, for any 𝑝>0, when 𝑡 approaches 𝑇 enough, 𝑃𝑆_𝑡≥𝑐𝜆_𝑡=exp−𝐼(𝑐)𝜆_𝑡+𝐻𝑎_𝑐√2𝜋𝑎_𝑐𝛽_𝑡×1+^𝑝_𝑘=1𝑏_𝑐,𝑘𝜆_𝑡1+𝑂𝜆_𝑡^𝑝+1, (4) where 𝜎²_𝑐=4𝑐²,𝛽_𝑡=𝜎_𝑐𝜆_𝑡,𝑎_𝑐=(1−2𝛼)²𝑐²−18𝑐²,𝐻𝑎_𝑐1=−2()𝑐log1−1−2𝛼2. (5) The coefficients 𝑏_𝑐,𝑘 may be explicitly computed as functions of the derivatives of 𝐿 and 𝐻 (defined in Lemma 3) at point 𝑎_𝑐. For example, 𝑏_𝑐,1 is given by 𝑏_𝑐,1=1𝜎²_𝑐−ℎ₂2−ℎ²₁2+𝑙₄8𝜎²_𝑐+𝑙₃ℎ₁2𝜎²_𝑐−5𝑙²₃24𝜎⁴_𝑐+ℎ₁𝑎_𝑐−𝑙₃2𝑎_𝑐𝜎²_𝑐−1𝑎²_𝑐, (6) with 𝑙_𝑘=𝐿^(𝑘)(𝑎_𝑐), and ℎ_𝑘=𝐻^(𝑘)(𝑎_𝑐).

2. Large Deviation for Energy

Given 𝛼>1/2, we first consider the following logarithmic moment generating function of 𝑆_𝑡; that is, 𝐿_𝑡(𝑢)∶=log𝔼_𝛼exp𝑢^𝑡₀𝑋²_𝑠(𝑠−𝑇)²𝑑𝑠,∀𝜆∈ℝ. (7) And let 𝒟_𝐿𝑡∶=𝑢∈ℝ,𝐿_𝑡(𝑢)<+∞ (8) be the effective domain of𝐿_𝑡. By the same method as in Zhao and Liu [5], we have the following lemma.

Lemma 3. Let 𝒟_𝐿 be the effective domain of the limit 𝐿 of 𝐿_𝑡; then for all 𝑢∈𝒟_𝐿, one has 𝐿_𝑡(𝑢)𝜆_𝑡(𝑢)+𝐻(𝑢)=𝐿𝜆_𝑡+𝑅(𝑢)𝜆_𝑡, (9) with 𝐿(𝑢)(𝑢)=−1−2𝛼−𝜑4,𝐻(𝜆)1=−21log2((𝑢,𝑅(𝑢)11+ℎ))=−2(𝑢)log1+1−ℎ(𝑢)(𝑢)𝜆1+ℎexp2𝜑_𝑡, (10) where 𝜑(𝑢)=−(1−2𝛼)²−8𝑢 and ℎ(𝑢)=(1−2𝛼)/𝜑(𝑢). Furthermore, the remainder 𝑅(𝑢) satisfies 𝑅(𝑢)=𝑂_𝑡→𝑇exp2𝜑(𝑢)𝜆_𝑡. (11)

Proof. By Itô’s formula and Girsanov’s formula (see Jacob and Shiryaev [11]), for all 𝑢∈𝒟_𝐿 and 𝑡∈[0,𝑇), log𝑑𝑃_𝛼𝑑𝑃_𝛽|_[0,𝑡]=𝛼−𝛽^𝑡₀𝑋_𝑠𝑠−𝑇𝑑𝑋_𝑠−𝛼²−𝛽²2^𝑡₀𝑋²_𝑠(𝑠−𝑇)²𝑑𝑠,_𝑡0𝑋_𝑠𝑠−𝑇𝑑𝑋_𝑠=12𝑋²_𝑡()+𝑡−𝑇^𝑡₀𝑋²_𝑠(𝑠−𝑇)²𝑡𝑑𝑠−log1−𝑇. (12) Therefore, 𝐿_𝑡(𝑢)=log𝔼_𝛽exp𝑢^𝑡₀𝑋²_𝑠(𝑠−𝑇)²𝑑𝑠𝑑𝑃_𝛼𝑑𝑃_𝛽|_[0,𝑡]=log𝔼_𝛽exp𝛼−𝛽2()𝑋𝑡−𝑇^2𝑡−𝛼−𝛽2𝑡log1−𝑇+12𝛽²−𝛼²×+𝛼−𝛽+2𝑢_𝑡0𝑋²_𝑠(𝑠−𝑇)²𝑑𝑠. (13) If 4𝑢≤(1−2𝛼)², we can choose 𝛽 such that (𝛽−1/2)²−(𝛼−1/2)²+2𝑢=0. Then 𝐿_𝑡(𝑢)(𝜆)=−1−2𝛼−𝜑4𝜆_𝑡−121log2((𝑢−11+ℎ))2(𝑢)log1+1−ℎ(𝑢)(𝑢)𝜆1+ℎexp2𝜑_𝑡, (14) where 𝜑(𝑢)=−(1−2𝛼)²−8𝑢, and ℎ(𝑢)=(1−2𝛼)/𝜑(𝑢). Therefore, 𝐿_𝑡(𝑢)𝜆_𝑡(𝑢)=−1−2𝛼−𝜑4−12𝜆_𝑡1log2((𝑢−11+ℎ))2𝜆_𝑡(𝑢)log1+1−ℎ(𝑢)(𝑢)𝜆1+ℎexp2𝜑_𝑡(𝑢)+𝐻(𝑢)=𝐿𝜆_𝑡+𝑅(𝑢)𝜆_𝑡. (15)

Proof of Theorem 1. From Lemma 3, we have 𝐿(𝑢)=lim_𝑡→𝑇𝐿_𝑡(𝑢)𝜆_𝑡=1−2𝛼−𝜑(𝑢)4, (16) and 𝐿(⋅) is steep; by the Gärtner-Ellis theorem (Dembo and Zeitouni [12]), 𝑆_𝑡/𝜆_𝑡 satisfies the large deviation principle with speed 𝜆_𝑡 and good rate function 𝐼(⋅) defined by the following: 𝐼(𝑥)=1))8𝑥((2𝛼−1𝑥−1²,if𝑥>0;+∞,if+𝑥≤0. (17)

Remark 4. Theorem 1 can also be obtained by using Theorem 1 in Zhao and Liu [5].

3. Sharp Large Deviation for Energy

For 𝑐>1/(2𝛼−1), let 𝑎_𝑐=(1−2𝛼)²𝑐²−18𝑐²,𝜎²_𝑐=𝐿^′′𝑎_𝑐=4𝑐³,𝐻𝑎_𝑐1=−2log(1−(1−2𝛼)𝑐). (18) Then 𝑃𝑆_𝑡≥𝑐𝜆_𝑡=_{𝑆𝑡≥𝑐𝜆𝑡}exp𝐿𝑎_𝑐−𝑐𝑎_𝑐𝜆_𝑡+𝑐𝑎_𝑐𝜆_𝑡−𝑎_𝑐𝑆_𝑡𝑑𝑄_𝑡=exp𝐿𝑎_𝑐−𝑐𝑎_𝑐𝜆_𝑡𝔼_𝑄exp−𝑎_𝑐𝛽_𝑡𝑈_𝑡𝐼_{{𝑈𝑡≥0}}=𝐴_𝑡𝐵_𝑡, (19) where 𝔼_𝑄 is the expectation after the change of measure 𝑑𝑄_𝑡𝑎𝑑𝑃=exp_𝑐𝑆_𝑡−𝐿_𝑡𝑎_𝑐,𝑈_𝑡=𝑆_𝑡−𝑐𝜆_𝑡𝛽_𝑡,𝛽_𝑡=𝜎_𝑐𝜆_𝑡. (20)

By Lemma 3, we have the following expression of 𝐴_𝑡.

Lemma 5. For all 𝑐>1/(2𝛼−1), when 𝑡 approaches 𝑇 enough, 𝐴_𝑡=exp−𝐼(𝑐)𝜆_𝑡+𝐻𝑎_𝑐1+𝑂(𝑇−𝑡)^𝑐. (21) For 𝐵_𝑡, one gets the following.

Lemma 6. For all 𝑐>1/(2𝛼−1), the distribution of 𝑈_𝑡 under 𝑄_𝑡 converges to 𝑁(0,1) distribution. Furthermore, there exists a sequence 𝜓_𝑘 such that, for any 𝑝>0 when 𝑡 approaches 𝑇 enough, 𝐵_𝑡=1𝑎_𝑐𝜎_𝑐2𝜋𝜆_𝑡1+^𝑝_𝑘=1𝜓_𝑘𝜆^𝑘_𝑡+𝑂𝜆_𝑡^−(𝑝+1). (22)

Proof of Theorem 2. The theorem follows from Lemma 5 and Lemma 6.

It only remains to prove Lemma 6. Let Φ_𝑡(⋅) be the characteristic function of 𝑈_𝑡 under 𝑄_𝑡; then we have the following.

Lemma 7. When𝑡approaches𝑇,Φ_𝑡belongs to 𝐿²(ℝ) and, for all𝑢∈ℝ, Φ_𝑡(𝑢)=exp−𝑖𝑢𝜆_𝑡𝑐𝜎_𝑐×exp𝐿_𝑡𝑎_𝑐+𝑖𝑢𝛽_𝑡−𝐿_𝑡𝑎_𝑐. (23) Moreover, 𝐵_𝑡=𝔼_𝑄exp−𝑎_𝑐𝛽_𝑡𝑈_𝑡𝐼_{{𝑈𝑡≥0}}=𝐶_𝑡+𝐷_𝑡, (24) with 𝐶_𝑡=12𝜋𝑎_𝑐𝛽_{𝑡|𝑢|≤𝑠𝑡}1+𝑖𝑢𝑎_𝑐𝛽_𝑡⁻¹Φ_𝑡(𝑢)𝐷𝑑𝑢,_𝑡=12𝜋𝑎_𝑐𝛽_{𝑡|𝑢|>𝑠𝑡}1+𝑖𝑢𝑎_𝑐𝛽_𝑡⁻¹Φ_𝑡(𝑢)𝐷𝑑𝑢,_𝑡=𝑂exp−𝐷𝜆_𝑡^1/3, (25) where 𝑠_𝑡𝑇=𝑠log𝑇−𝑡^1/6, (26) for some positive constant𝑠, and𝐷is some positive constant.

Proof. For any𝑢∈ℝ, Φ_𝑡(𝑢)=𝔼exp𝑖𝑢𝑈_𝑡exp𝑎_𝑐𝑆_𝑡−𝐿_𝑡𝑎_𝑐=exp−𝑖𝑢𝜆_𝑡𝑐𝜎_𝑐×exp𝐿_𝑡𝑎_𝑐+𝑖𝑢𝛽_𝑡−𝐿_𝑡𝑎_𝑐. (27) By the same method as in the proof of Lemma 2.2 in [7] by Bercu and Rouault, there exist two positive constants𝜏and𝜅such that Φ_𝑡(𝑢)²≤1+𝜏𝑢²𝜆_𝑡^{−(𝜅/2)𝜆𝑡}; (28) therefore, Φ_𝑡(⋅)belongs to𝐿²(ℝ), and by Parseval’s formula, for some positive constant𝑠, let 𝑠_𝑡𝑇=𝑠log𝑇−𝑡^1/6; (29) we get 𝐵_𝑡=12𝜋𝑎_𝑐𝛽_{𝑡|𝑢|≤𝑠𝑡}1+𝑖𝑢𝑎_𝑐𝛽_𝑡⁻¹Φ_𝑡(𝑢)1𝑑𝑢+2𝜋𝑎_𝑐𝛽_𝑡×_{|𝑢|>𝑠𝑡}1+𝑖𝑢𝑎_𝑐𝛽_𝑡⁻¹Φ_𝑡(𝑢)𝑑𝑢=∶𝐶_𝑡+𝐷_𝑡,𝐷_𝑡=𝑂exp−𝐷𝜆_𝑡^1/3, (30)(31)(32) where𝐷is some positive constant.

Proof of Lemma 6. By Lemma 3, we have 𝐿_𝑡^(𝑘)𝑎_𝑐𝜆_𝑡=𝐿^(𝑘)𝑎_𝑐+𝐻^(𝑘)𝑎_𝑐𝜆_𝑡+𝑂𝜆^𝑘_𝑡(𝑇−𝑡)^−2𝑐𝜆_𝑡. (33) Noting that𝐿^′(𝑎_𝑐)=0, 𝐿^′′(𝑎_𝑐)=𝜎²_𝑐 and 𝐿^′′𝑎_𝑐2𝑖𝑢𝛽_𝑡²𝜆_𝑡𝑢=−²2, (34) for any𝑝>0, by Taylor expansion, we obtain logΦ_𝑡(𝑢)𝑢=−²2+𝜆_𝑡^2𝑝+3_𝑘=3𝑖𝑢𝛽_𝑡^𝑘𝐿^(𝑘)𝑎_𝑐+𝑘!_{2𝑝+1𝑘=1}𝑖𝑢𝛽_𝑡^𝑘𝐻^(𝑘)𝑎_𝑐|𝑢|𝑘!+𝑂max1,^2𝑝+4𝜆_𝑡^𝑝+1; (35) therefore, there exist integers𝑞(𝑝), 𝑟(𝑝)and a sequence𝜑_𝑘,𝑙independent of  𝑝; when𝑡approaches𝑇, we get Φ_𝑡(𝑢)−𝑢=exp²211+𝜆_𝑡^{2𝑝𝑘=0𝑞(𝑝)𝑙=𝑘+1}𝜑_𝑘,𝑙𝑢^𝑙𝜆_𝑡^𝑘/2|𝑢|+𝑂max1,^𝑟(𝑝)𝜆_𝑡^𝑝+1, (36) where𝑂is uniform as soon as|𝑢|≤𝑠_𝑡.
Finally, we get the proof of Lemma 6 by Lemma 7 together with standard calculations on the𝑁(0,1)distribution.

Acknowledgment

This research was supported by the National Natural Science of Tianyuan Foundation under Grant 11226202.