On the Exponentials of Fractional Ornstein-Uhlenbeck Processes

We study the correlation decay and the expected maximal increment (Burkholder-Davis-Gundy type inequalities) of the exponential process determined by a fractional Ornstein-Uhlenbeck process. The method is to apply integration by parts formula on integral representations of fractional Ornstein-Uhlenbeck processes, and also to use Slepian’s inequality. As an application, we attempt Kahane’s T-martingale theory based on our exponential process which is shown to be of long memory.


Introduction
We begin with a review on the definition and properties of fractional Brownian motion (FBM for short). It is well known that FBM has stationary increments and self-similarity with index H, i.e., for any Another important property is long memory. Although there exist several notions, we use the following definition which is widely known.
Note that E[(X 1 ) 2 ] < ∞ implies that the correlation and the covariance are different only in multiplication of some constant and we use the latter here. The definition is related to the classical invariance principle, i.e., if ∞ n=0 |Γ(n)| < ∞, the properly normalized sum of (X 1 , X 2 , . . .) converges weakly to Brownian motion (BM for short). By contrast, if the process has a long memory, this may not hold (see p.191 of [18] or p.336 of [19]). For other definitions and their relations one can consult [18]. Thus ∞ n=0 Γ h (nh) = ∞ and the incremental process of FBM with H ∈ ( 1 2 , 1) is proved to have a long memory property. Concerning the maximal inequality of FBM there have been several results; see, for example, Chapter 4.4 of [5]. In particular [13] proved the maximal inequality for FBM with H ∈ ( 1 2 , 1) which corresponds to the Burkholder-Davis-Gundy inequality for martingale: For p > 0 there exist constants c(p, H) and C(p, H) such that where τ is a stopping time. Extending this, [11] have obtained inequalities for the moment of integrals with respect to FBM. Regarding other properties, we refer to [5] or [19] which give conclusive introduction to FBM. We also refer to recently published [12] which gives nice summary on stochastic calculus for FBM. Now we turn to Ornstein-Uhlenbeck processes driven by FBM with H ∈ (0, 1 2 ) ∪ ( 1 2 , 1) (FOU for short), which is defined by where λ > 0, σ > 0 and ξ is a.s. finite random variable. This process appears firstly in [4]. They show that FOU {Y H,ξ t } t≥0 is the unique a.s. continuous-path process which solves and is strictly stationary if We mainly study this stationary version, as follows where the random function t → Y H t now can be and will be extended to the whole t ∈ . Let 2π a constant. The correlation decay of {Y H t } t∈ with H ∈ (0, 1 2 ) ∪ ( 1 2 , 1) as s → ∞ satisfies, see p.289 of [15],  [17] more generally obtained estimates of the tail of the maximum of stochastic integrals with respect to FBM, of which we shall make use. Other interesting results are given in Chapters 1.9 and 1.10 of [12].
In recent years, it has been of great interest to study the exponential functionals and the exponential processes determined by BM and Lévy processes, see [2] and [3], with the view toward application in financial economics. In this paper, we study the exponential process determined by {Y H t } t∈ , We shall call the process to be a geometric fractional Ornstein-Uhlenbeck processes (gFOU, for short). We study two fundamentally important properties of gFOUs. The correlation decay Cov X H t , X H t+s as s → ∞, and the expected maximal increments The first result is useful to understand the spectral structure of the process. The second result is of intrinsic importance to the path variation (and hence toward various applications) of the process. In case {Y H t } with H = 1/2 (BM case), some weaker form of the results appears very recently in a paper by Anh, Leonenko and Shieh (2007), whose methods are based on Hermite orthogonal expansion and the Itô's calculus for martingales. However, both tools are lack for FBM case, since FBM is not a semimartingale. Thus we need to use other devices, which are mainly precise calculations based on the Gaussian properties, the integral representations of FOUs, and the Slepian's inequality. We remark that the main results Theorems 2.1 and 2.2 in this paper are new even in the BM case, to our knowledge.
This paper is organized as follows. In Section 2 we state the main results. In Section 3 we treat Kahane's T-martingale theory as an application. We present all proofs of our results in Section 4.

The main results
From now on we treat gFOU and FOU with λ = σ = 1 for convenience. Moreover, we consider these process on . The notation d = denotes equality in distributional sense, for processes also for random variables or vectors. All proofs of our results are given in the final section.
As a preliminary step we confirm the following basic result.  3 4 ) no longer has the long memory property. We also observe that the situation depends entirely on m being even or being odd, rather than the actual value of m. Now we turn to the more difficult second part, namely to study expected maximal increment of gFOU. For the upper bound inequality, we only consider those gFOUs with H ∈ ( 1 2 , 1), and we are not able to obtain the case of H < 1/2, yet this latter case is of less interest in view that the process if not of long range dependence then. Before analyzing, we present three lemmas which we think themselves to be interesting in future researches. Indeed Lemmas 2.3 and 2.4 deal with maximal inequalities for FOUs. For the consistency, in all the following statements, we always include the H = 1/2 case.
The first result (Lemma 2.3) is based on Statement 4.8 of [17]. It will be useful to give a clean statement of this since the definition of FBM is different from ours and there are minor mistakes in [17] (e.g., regarding his q f (s, t) a constant is lacking, he referred to Theorem 4.1 in Statement 4.2 but we can not find Theorem 4.1 in his paper). 1). Then for any λ ≥ 0, r ≥ 0 and t ∈ we have where c 1 is an universal constant, which does not depend on any m or H.

Lemma 2.5.
Let H ∈ (0, 1) and p > 0. Then FOU {Y H t } t∈ has the following lower bound for p-th moment of maximal increments for all 0 ≤ r ≤ T and all t ∈ .
where c 2

(p, T, H) is a constant depending on parameters p, T and H.
Now we state our main results. The upper bound inequality is given as follows.

An application to Kahane's T-martingale Theory
J.-P. Kahane established T-martingale Theory as a mathematical formulation of Mandelbrot's turbulence cascades; see [8] and [14] for inspiring surveys. To our knowledge, the theory is only applied to independent or Markovian cascades. Using the results in Section 2 we are able to give an dependent attempt to this theory. To describe our result, let X be a normalized gFOU which is defined to be, for a given fixed H ∈ ( 1 2 , 1) and its corresponding stationary FOU Y H t , where c H is chosen so that the resulting positive-valued stationary process is of mean 1. Note that the process X is non-Markovian (indeed, it is of long range dependence). Now let a sequence of independent gFOU X n , defined on a common probability space (Ω, P); each process {X n (t)} t∈ is of continuous paths and is distributed as {X (b n · t)} t∈ , n = 0, 1, 2, . . . , where the scaling factor We consider the integrated process of the n + 1 products, We note that, for each t, A n (t, ω) is well-defined as an integral for path-wise ω, since the integrand is a positive-valued continuous function in s for path-wise ω. The following two facts are basic to the theory: 1. for each t fixed, the sequence A n (t) form a martingale in n.
2. for each n fixed, t → A n (t) is continuous and increasing. We state our T-martingale result for the gFOU process as follows. In the statement, we restrict the time parameter for the target process A(t) to be A(t), t ∈ [0, 1]; though it can be defined for any compact time-interval [0, T ].
Remark 3.1. The close form of E(X (0)) q can be written out, since the random variable Y H t is Gaussian distributed; it is non-linear in q, which is the heart of the matter. 's theory, the initial process (in our case, X ) is assumed to be independent (for discrete cascades) or to be Markovian (for general cascades). The above result can be regarded to be a first attempt to apply the dependent process X to Kahane's theory, which theory aims to proceed some multi-scale analysis (usually termed as "multifractal analysis") on the atomless random measure induced by the continuously increasing process A.

Remark 3.3.
We mention that, in [10] the authors adapt Kahane's formulation to stationary processes, and impose various conditions to enforce the validity of their re-formulation (in their eventual examples, one is a two-state Markov process and one is a Poisson process with random magnitudes). It has been a recent study to examine the validity of their re-formulation for several stationary exponential processes, see [1] and the references therein.

The proofs
Proof of Lemma 2.1 It follows that for any real set (t 1 , t 2 , . . . , t n ) and all h ∈ , Accordingly our assertion is implied by

By aid of the equation (3), the equation above can be written as
Cov Hence we obtain the result.

Proof of Proposition 2.2 As in Proof of Proposition 2.1, (Y H t , Y H t+s ) is a bivariate Gaussian distribution and its moment generating function writes
Here we use the expansion of e x and the multinomial expansion. By using this representation we and hence the remaining term of the sum in (9) is only that of n = m, Then putting m = 2k + l = 2(m − l − k) + l, we have Here we use the formula (2n − 1)!! = (2n)!/(2 n n!). When m is odd only terms l = 1, 3, 5, . . . , m remain and it follows form the equation (3) that When m is even only terms l = 2, 4, . . . , m remain and it follows from the equation (3) that .
Finally by the reflection principle for Gaussian Markov processes (e.g. [16]), it follows that Second the symmetric property of FBM gives From this we have Thus we get the second assertion. Next the self-similarity of FBM and Since for positive random variable X , we have Finally, we apply relations 2 and 3 of 3.461 of [6], i.e.,  Take absolute value of this to obtain Then taking expectation of maximum of this, we have On behalf of Lemma 2.3, the expectation in each term of the sum is bounded as Here M Note that (1 − e −s )/r H with 0 ≤ s ≤ r and H ∈ [ 1 2 , 1), is uniformly bounded in r ≥ 0. In addition c m / m! with c > 0 is also uniformly bounded in m. Hence we can take a universal constant c 1 > 0 and obtain Proof of Lemma 2. 5 We have easily Since {Y H t } t∈ is a Gaussian process, a random vector (Y H 0 , Y H r ) is a bivariate Gaussian distribution and its linear combination (Y H r − Y H 0 ) is also univariate Gaussian. We denote the square root of variance asσ and then Here we use the equality ∞ 0 x ν−1 e −µx p d x = 1 p µ −ν/p Γ(ν/p), Re µ > 0, Re ν > 0, p > 0 from 3.478 of [6]. An evaluation ofσ p is derived via the equation (2) with σ = λ = 1.
Accordingly it follows thatσ p ≥ (8c H c T ) p/2 r pH .
Substituting this into (11) By virtue of Lemma 2.4, it follows that

Proof of Proposition 3.1
The proof is based on the examination of the validity of several crucial assumptions imposed in [10], in which the authors adapt Kahane's formulation to stationary processes, Our results in Section 2 assert that the crucial conditions imposed in their paper hold for our exponential process X (t). Firstly, our Proposition 2.1 on the decay of the covariance function asserts the sufficiency conditions in their subsections 3.1 and 3.2 hold for X . Namely, we apply our Proposition 2.1 to see that the L 2 (d P) norm of the martingale difference, ||A n (1) − A n−1 (1)|| 2 , is summable in n (their subsection 3.1), and thus a limiting process A(t) exists, as the L 2 (d P) limit of A n (t) for each t. While it is obvious that A(t) is path-wise increasing in t, we need apply again our Proposition 2.1 to see that it is indeed path-wise non-degenerate and continuous in t (their subsection 3.2). Secondly, our Theorem 2.1 asserts that, for all q ∈ [1, 2], where C(H) is a constant derived from Theorem 2.1 and the normalizing factor defined in X (t).
Since we have a dominating convergent series on the right-handed side of the above display, the main (and most crucial) assumption (10) in their Proposition 5 of subsection 3.3 indeed holds, and hence the inequality of Proposition 3.1 is established as a consequence of their Proposition 5 of subsection 3.3.
Concluding remarks on some future works: 1. We may try to obtain the sharp bounds for C(H) and c(H) in Theorems 2.1 and 2.2. This will be a significant supplement to the works in [13] (in which some rather sharp bounds for B-D-G