Large Deviation Principle for Fractional Brownian Motion with Respect to Capacity

We show that the fractional Brownian motion (fBM) defined via the Volterra integral representation with Hurst parameter H≥12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H\geq \frac {1}{2}$\end{document} is a quasi-surely defined Wiener functional on the classical Wiener space, and we establish the large deviation principle (LDP) for such an fBM with respect to (p,r)-capacity on the classical Wiener space in Malliavin’s sense.


Introduction
Quasi-sure analysis, as a powerful tool to study functions on infinite dimensional spaces, was initiated by Malliavin [33][34][35] using the stochastic calculus of variations and Fukushima [16] by means of Dirichlet forms. The theory can be applied to plentiful aspects in stochastic analysis, such as Markov processes, Gaussian processes, large deviation principles and etc., see e.g. [1, 4, 8, 11, 15, 17-19, 25, 30, 42, 43, 45] and the literature there-in. Malliavin observed that by constructing a regularity theory and a uniform measure on an abstract Wiener space, many interesting Wiener functionals are smooth, and this regularity theory enables us to study Wiener functionals like in the finite dimensional real analysis. An outer measure called -capacity, denoted by throughout this paper, was introduced in terms of the Malliavin derivative and a number of papers concerning capacities have been published throughout last decades, see e.g. [15,[25][26][27][35][36][37] and the literature there-in.
Among results related to quasi-sure analysis, the majority considers solutions to Itô's stochastic differential equations, which are merely measurable Wiener functionals. In fact, quasi-sure analysis can be used to handle processes which are measurable functionals on the Wiener space but not solutions to any Itô's SDEs, such as fractional Brownian motions. Fractional Brownian motion (fBM), as an important example of Gaussian processes, has a variety of applications in mathematical finance, hydrodynamics, communication networks and so on, see e.g. Biagini et al. [3], Decreusefond andÜstünel [6], Mandelbrot and Van Ness [38], Mishura [40] and etc. To be more precise, an fBM 0 with Hurst parameter 0 1 is a centred self-similar Gaussian process, whose covariance function is given by .
By definition, one obtains immediately that when 1 2 , this process is a standard Brownian motion. However, when takes other values, this process differs from Brownian motion as its increments are no longer independent and thus exhibits long memory behaviour. Thanks to the Volterra integral representation introduced by Decreusefond andÜstünel [6], which is 0 0 where 0 is a standard Brownian motion and is some singular kernel, fBMs can be regarded as measurable Wiener functionals. According to Malliavin, fBMs with different Hurst parameters induce a family of capacities living on distinct abstract Wiener spaces. Nonetheless, all these fBMs can be viewed as Wiener functionals on the classical Wiener space due to the integral representation, so that we can study them with one uniform measure -the capacity associated with Brownian motion. In this paper, we will prove that the integral representation of fBM is defined except for a capacity zero set when Hurst parameter 1 2 , that is, fBMs are quasi-surely defined Wiener functionals on the classical Wiener space.
In order to achieve this goal, we need several results from the rough paths theory. The analysis of rough paths, originated by Lyons (see e.g. [13,14,31,32]), was established to study solutions to stochastic differential equations driven by semi-martingales and other rough signals. It turns out that many techniques developed in the rough paths analysis can be applied in the research of quasi-sure analysis as demonstrated in [4], as well as a series of work by various researchers (see e.g. [21-24, 28, 39] and etc.).
Besides proving that fBMs as Wiener functionals are quasi-surely defined, we establish large deviation principles for these Wiener functionals. Large deviations theory has been a prevalent topic in probability for its significance in statistics and statistical mechanics. It completes the central limit theorem by telling us that tail probabilities decay exponentially fast. In 1970s, the theory of large deviation principles (LDP for short) experienced rapid development due to the remarkable work by Donsker and Varadhan [10], and one may refer to [7,9,10,44] for further details. In finite dimensional case, one crucial result in this theory is the Cramér's theorem, which precisely describes the rate of exponential decay. In infinite dimensional case, the exponential decay of large perturbations of Brownian motion from its mean trajectory is characterised in the Schilder's theorem, and the Freidlin-Wentzell theorem generalises it to the laws of Itô diffusions, see e.g. [7] and [9]. Similar results were proved using rough paths theory by Ledoux, Qian and Zhang in [28], see also [12,39]. Moreover, the general Cramér's theorem (see e.g. [9]) can be used to study the large deviations of Gaussian measures. Indeed, large deviation principles can be formulated for not only measures, but also capacities. In [45], Yoshida established a version of LDP with respect to capacities on the abstract Wiener space, which implies the LDP for Gaussian measures, while in [17] and [18], Gao and Ren considered the capacity version of Freidlin-Wentzell theorem. This line of research was taken a step further to the setting of Gaussian rough paths in [4]. Inspired by the arguments in [4], we prove the LDP for fBMs with respect to -capacity on the classical Wiener space. The major difference between this paper and [45] is that we use different capacities. Although the invariance property of capacities has been proved in [2], the capacities associated with different Gaussian measures are however non-comparable. Therefore, instead of using the capacities induced by fBMs, we treat fBMs with different Hurst parameters as a family of Wiener functionals and choose the Brownian motion capacity as a uniform measure.
This paper is organised in the following way. In next section, we present a few definitions and properties of capacities, and then we state the main result, a quasi-sure version of large deviation principle for fractional Brownian motions realised as Wiener functionals on the classical Wiener space. In Section 3, we recall several elementary results such as Wiener chaos decomposition, exponential tightness, contraction principle in the context of quasisure analysis. Then in Section 4, with a quite technical proof, we provide a construction of quasi-surely defined modifications of fBMs, which are considered to be Wiener functionals in our settings. Next, in Section 5, we prove the exponential tightness of the family of finite dimensional approximations of fBMs (modified as in Section 4). In this section, we adopt several ideas from the rough paths theory. Finally, we determine the rate function and complete our proof of the quasi-sure large deviation principles for fBMs in Section 6. The key step is to obtain the finite dimensional quasi-sure large deviation principles, which may be accomplished by explicit computations.

Preliminaries and the Main Result
In this section, we will introduce basic definitions in Malliavin calculus and state the main result.

Malliavin Differentiation and Capacities
We mainly follow the notations used in Ikeda and Watanabe [20], and Nualart [41]. Although our presentation applies to multi-dimensional case as well, we only consider the one dimensional Wiener space here for simplicity. Let be the space of all real-valued continuous paths over time interval 0 1 starting from the origin, equipped with the uniform norm given by sup 0 1 , . The Borel -algebra is denoted by B . We call the functions which send each to its coordinates (where 0) the coordinate mapping processes, and are denoted by or . The Wiener measure P is the distribution of standard Brownian motion, the unique probability on B such that 0 is a standard Brownian motion. Let F be the completion of B under P. The Wiener functionals, by convention in the literature, are F -measurable functions on .
Let H denote the Cameron-Martin space, which is a Hilbert space containing all absolutely continuous functions on 0 1 such that 0 0 and its generalised derivative is square integrable. The inner product on H is given by for any F .
The first Borel-Cantelli lemma can be proved in the context of capacities, and it says that if a sequence of subsets 1 of satisfies 1 , then lim sup 0. One may refer to [35] for a proof. Another important tool used in this paper is the capacity version of Chebyshev's inequality, which is D for every lower semi-continuous D and 0.

The Main Result
We are now ready to introduce the definition of large deviation principle (LDP) with respect to capacities as in [4] and [45]. Then we define fractional Brownian motions and its integral representation according to [6] (see also Chapter 5 in [41]) and state the main result at the end of this section. Recall that a fractional Brownian motion (fBM) 0 with Hurst parameter 0 1 is a self-similar Gaussian process with stationary increments whose covariance function is given by

Definition 1 Let
FBMs can be realised as Wiener functionals in the way that where 0 is the coordinate mapping process on (hence a Brownian motion). is defined almost surely -the integral on the right-hand side is understood in the Itô's sense. is a singular kernel, and when 1 2 , it is given by where is some constant depending only on . It is straightforward by the Kolmogorov continuity theorem that the process defined by Eq. 1 has a modification which is -Hölder continuous with . We abuse our notation by using 0 to denote such a modification, and in the sequel, when we mention fBM, we always refer to this modified version.
To state the quasi-sure large deviation principles for fBMs, we need to identify the corresponding rate functions, which must be the same rate functions as in the context of probabilities. Notice that (1) defines a mapping taking almost all Brownian motion paths to the paths of fBM with Hurst parameter . Let Q P 1 be the pushforward of the Wiener measure, which is the distribution of this version of fBM, a Gaussian measure on B . Similar to the case of Brownian motion, there is a canonical way to associate this Gaussian measure and with a separable Hilbert space H , which can be continuously and densely embedded into (see e.g. [5]). In [6], the Cameron-Martin space H corresponding to the fractional Brownian motion with Hurst parameter was identified, and is the space consisting of all elements of the form 0 , where is as in Eq. 2, and 2 0 1 . The inner product on H is defined as Let Q be the family of scaled measures, the laws of under Q. By definition for each B . According to Theorem 3.4.12 on page 88 in [9], Q satisfies the LDP with the good rate function given by (1) There exists a modification of (for 0) defined -quasi-surely.
The remaining of this paper is devoted to the proof of the above result, Theorem 2. The first part of Theorem 2 follows from Theorem 8 in Section 4 directly. The proof of the second part will be presented in Sections 5 and 6.
Our strategy of the proof is the following. For each fixed 0 1 and N, we consider , a finite linear combination of elements in the classical Wiener space, and show that the sequence N converges quasi-surely (with respect to the Brownian motion capacity). The main difficulty here is that the kernel is singular in , so it is very difficult to control its increments in and estimate the integral of over small time intervals near time 0. However, we notice that as a function of , behaves more regularly. Therefore, we control the difference by the difference of when varies. Then we may obtain the desired mapping , a quasi-surely defined modification of fBM on the Wiener space by approximations with linear interpolations. Define For each , is quasi-surely defined, then we show that this sequence N converges to some mapping quasi-surely. Since any countable union of capacity zero sets still has zero capacity, we conclude that the limit is quasi-surely defined, and we shall also see that this convergence is exponentially fast in the proof. Now as the large deviation principle may be established for 's, using exponentially good approximations result from the LDP theory, we deduce the -LDP for the limit mapping .

Some Technical Facts
In this section we collect a few technical facts which will be used to prove the quasi-sure version of large deviation principles for fBMs. where G is the -algebra generated by random variables H (see e.g. Theorem 1.1.1, Section 1.1 in [41] and its proof). The projection from 2 to H is denoted by . Let P 0 denote the space of polynomial random variables of the form 1 1 H where is a polynomial with degree less than or equal to , and let P be the closure of P 0 in 2 . Then it follows that for every integer , P 0 H . If D for some 2, then for all ,

Wiener Chaos Decomposition
For a proof of this relation, one may refer to Section 1.2 in [41].
for any 2 and .

Exponential Tightness
Most conclusions in the theory of large deviations (see [7,9] for details) are still valid in the context of capacities. Let us state some of them which will be used in this paper, their proofs are routine and will be omitted.

Proposition 4 (Varadhan's Contraction Principle) Let
0 be a family of quasi-surely defined maps from to a Polish space 1  To deal with fBMs, which are merely measurable Wiener functionals, the concept of exponential tightness is a useful technique in proving large deviation principles. The natural modification of this notion can be formulated as the following.
The following version of the contraction principle will be useful in our proof.

FBMs as Wiener Functionals
Let 0 the continuous version of fBM with Hurst parameter defined via (1). According to the transfer principle in [41] (see Proposition 5.2.1, Section 5.2, [41]), D , and its Malliavin derivative may be computed explicitly as in the following lemma (see also [29]).
Its higher order Malliavin derivatives vanish identically. The first part of Theorem 2 is a consequence of the following result. -quasi-surely to some limit, denoted by too, which is also the limit of Proof The proof is quite technical and will be divided into several steps. When 1 2 , an fBM is a standard Brownian motion, and hence the result follows immediately. We only need to consider the case when Since is increasing in and , it suffices to prove that the above infinite sum is finite for 2 and all N. Therefore, we shall assume that 2 in the sequel.
Step 1. In this step, we convert our problem from estimating the capacities to estimating the 2 -norm of Gaussian random variables. By Chebyshev's inequality, we have for any 0. Since 1 2 is a polynomial functional of degree 2, and 1 1 where is defined as in Eq. 12, so for all 3, 1 2 0.
Therefore, by Eq. 3, Since Brownian motion has independent increments, we have Step 2. In this step, we further simplify our problem using a rather simple observation. By change of variables, for each 0 2 1 , Using the definition of and change of variables, we observe that for all 0 , and hence .
On the other hand, for every 0 , Step 3. In this step, we find upper bounds for 2 and 2 , respectively. We first find a control of Then for all , 0 and 0. By Hölder's inequality, we obtain that We control the integral of first. When 1 2 , since 1, we have 1 2 i.e. 1 2 . Therefore, we deduce that (24) Step 4. In this step, we complete our proof using the above estimates. It follows from Eq. 19, 23 and 24 that and applying the first Borel-Cantelli lemma as before. If for , there are infinitely many 's such that 1 1, then N is not Cauchy. Therefore, by Chebyshev's inequality, As a consequence, N converges to apart from on a slim set, and the uniqueness of limit forces its limit to be , which implies q.s.
From now on, we work with the modification of which is the -quasi-sure limit of the approximations .

Exponential Tightness of the Approximation Sequence
For each fixed , is quasi-surely defined (with 0 0 for all ). We define a map by Then is -quasi-surely defined as it is a linear interpolation of finitely many 's for all and . For each , let be the scaled map, which is defined as . As is the limit of linear combinations of 's, it follows that . Our goal is to show that the sequence N converges to some quasi-surely, which implies that N converges to quasi-surely, where the scaled map is given by . Moreover, the fact that the sequence of scaled maps N converges exponentially fast will be revealed in the proof as well. Since is quasi-surely defined with exponentially good approximations N , we may apply the result from the LDP theory to conclude the final result.
We will need the following estimate from the rough path analysis, which is contained in [31] (see Proposition 4.1.1 on page 62 or equation (4.15) on page 64). Here, we adapt the result to our case and state it as the following: where the supremum is taken over all finite partitions of 0 1 , 2 for 1 2 , 0 2 , and is the increment of path .
Together with Proposition 3, the above estimate allows us to simplify our problem by controlling the 2 -norm of Gaussian processes instead of capacities.

Theorem 10 For
N and 1 , N converges -quasi-surely to some limit , and the scaled maps N are exponentially good approximations of 0 under the capacity .
Proof Here we use a technique in the theory of rough paths to control the tails of 's, which are Gaussian. Let us first prove that the sequence N converges uniformly -quasi-surely. By using the elementary fact that sup 1 1 1 for any and for any 1, where the supremum is taken over all possible finite partitions of 0 1 , and , together with Proposition 9, we obtain that where is a constant depending on and , and 2 . We will apply the above estimate to to obtain an upper bound of 1 (25) where 0. Since is increasing in , we shall assume that 2. By monotonicity and sub-additivity properties of capacity, we obtain that for 0, It thus implies that depends on , , and . Plugging (31) into (28), we obtain that Applying the same argument as in the previous theorem, we see that the problem may be reduced to proving that for a suitable positive 0, Then by the first Borel-Cantelli lemma for capacity, we obtain the quasi-sure convergence for N . Since Thus, the convergence of the series in Eq. 36 implies the convergence of N . Denote its limit by , then is defined quasi-surely on .
Next, we prove the sequence 1 0 converges to 0 exponentially fast with respect to the capacity , that is, lim lim sup 0 2 log .
To this end, we shall use a similar argument as in the proof above. By the sub-additivity of capacity, for 0, As shown in Theorem 10, 0 are exponentially good approximations of 0 , so it suffices to verify that the function defined above coincides with the function given in Eq. 9 and satisfies all conditions in Proposition 6. Let us first check if satisfies all conditions. We observe that given in Eq. 4 is a good rate function by definition.
For any closed , denote inf , where is defined as in Eq. 46. By definition, inf 1 . Suppose that lim inf , then as is a good rate function and lower semi-continuous functions attain their minimums on compact sets, we conclude that attains its minimum on the closed subset 1 . Therefore, for each , there exists some such that 1 and inf 1 .
We notice that for all , in as . Since for all , for each 0, for large , where . It follows that inf inf for sufficiently large, and hence by taking limit infimum on both sides, we deduce that inf lim inf inf .
According to Lemma 4.1.6 (a), Section 4.1.1 in [7], inf lim inf inf (47) when letting 0, and hence the condition (10) is fulfilled. The case when lim inf is trivial, so we have verified all conditions in Proposition 6. Next, we prove that coincides with the function defined as in Eq. 9 by sup 0 lim inf inf .
For any , set in Eq. 47. It holds that inf lim inf inf lim inf inf .
By letting 0 and applying Lemma 4.1.6 (a), Section 4.1.1 in [7], we conclude that for all . For the reverse part, denote for , and as shown above, as . Therefore, for any 0, there exists some 0 such that for all , inf .
By the definition of , . By taking limit infimum over first, then supremum over , we obtain that sup 0 lim inf inf and hence .
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