Gaussian and non-Gaussian processes of zero power variation
aa r X i v : . [ m a t h . P R ] M a y Gaussian and non-Gaussian processes of zero power variation
Francesco Russo ∗† and Frederi VIENS ‡ November 8, 2018
Abstract
This paper considers the class of stochastic processes X defined on [0 , T ] by X ( t ) = R T G ( t, s ) dM ( s ) where M is a square-integrable martingale and G is a deterministickernel. When M is Brownian motion, X is Gaussian, and the class includes fractional Brownianmotion and other Gaussian processes with or without homogeneous increments. Let m be anodd integer. Under the assumption that the quadratic variation [ M ] of M is differentiable with E [ | d [ M ] ( t ) /dt | m ] finite, it is shown that the m th power variationlim ε → ε − Z T ds ( X ( s + ε ) − X ( s )) m exists and is zero when a quantity δ ( r ) related to the variance of an increment of M over asmall interval of length r satisfies δ ( r ) = o (cid:0) r / (2 m ) (cid:1) .In the case of a Gaussian process with homogeneous increments, δ is X ’s canonical metric,the condition on δ is proved to be necessary, and the zero variation result is extended to non-integer symmetric powers, i.e. using | X ( s + ε ) − X ( s ) | m sgn ( X ( s + ε ) − X ( s )) for any realvalue m ≥
1. In the non-homogeneous Gaussian case, when m = 3, the symmetric (generalizedStratonovich) integral is defined, proved to exist, and its Itˆo formula is proved to hold for allfunctions of class C . KEY WORDS AND PHRASES : Power variation, martingale Volterra convolution, co-variation, calculus via regularization, Gaussian processes, generalized Stratonovich integral, non-Gaussian processes.
MSC Classification 2000 : 60G07; 60G15; 60G48; 60H05.
The purpose of this article is to study wide classes of processes with zero cubic variation, and moregenerally, zero variation of any order. Before summarizing our results, we give a brief historicaldescription of the topic of p -variations, as a basis for our motivations. ∗ ENSTA-ParisTech. Unit´e de Math´ematiques appliqu´ees, 32, Boulevard Victor, F-75739 Paris Cedex 15 (France) † INRIA Rocquencourt Projet MathFi and Cermics Ecole des Ponts ‡ Department of Statistics, Purdue University, 150 N. University St., West Lafayette, IN 47907-2067, USA .1 Historical background The p -variation of a function f : [0 , T ] → R is the supremum over all the possible partitions { t < . . . < t N = T } of [0 , T ] of the quantity N − X i =1 | f ( t i +1 ) − f ( t i ) | p . (1)The analytic monograph [6] contains an interesting study on this concept, showing that a p -variationfunction is the composition of an increasing function and a H¨older-continuous function. The notionof p -variation of a stochastic process or of a function was rediscovered in stochastic analysis, partic-ularly in the context of pathwise (or quasi-pathwise) stochastic calculus. The fundamental paper[11], due to H. F¨ollmer, treats the case of 2-variations. More recent dealings with p -variations andtheir stochastic applications, particularly to rough path and other integration techniques, are de-scribed at length for instance in the books [7] and [16] , which also contain excellent bibliographieson the subject.For p = 2, the Itˆo stochastic calculus for semimartingales has mimicked the notion of 2-variation,with the notion of quadratic variation. Let S be a semimartingale; as in (1), consider the expression N − X i =1 | S ( t i +1 ) − S ( t i ) | . (2)One defines the quadratic variation [ S ] of S as the limit in probability of the expression in (2) asthe partition mesh goes to 0, instead of considering the supremum over all partitions. Moreover,the notion becomes stochastic. In fact for a standard Brownian motion B , its 2-variation [ B ] is a.s.infinite, but its quadratic variation is equal to T . In order to reconcile 2-variations with the finite-ness of [ B ], many authors have proposed restricting the supremum in (1) to the dyadic partitions.However, in Itˆo calculus, the idea of quadratic variation is associated with the notion of covariation(also known as joint quadratic variation) [ S , S ] of two semimartingales S , S , something whichis not present in analytic treatments of p -variation. This covariation [ S , S ] is obtained by polar-ization of (2), i.e. is the limit in probability of P N − i =1 (cid:0) S ( t i +1 ) − S ( t i ) (cid:1) (cid:0) S ( t i +1 ) − S ( t i ) (cid:1) when,again, the partition mesh goes to zero.In the study of stochastic processes, the p -variation has been analyzed in some specific cases,such as local time processes (see [24]), iterated Brownian motion, whose 4-th variation is finite, andmore recently fractional Brownian motion (fBm) and related processes. To work with a generalclass of processes, the tools of Itˆo calculus would nonetheless restrict the study of covariationto semimartingales. In [27], the authors enlarged the notion of covariation to general processes X and Y . They achieved this by modifying the definition, considering regularizations instead ofdiscretizations. One starting observation is the following. Let f : [0 , T ] → R be continuous. This f has finite variation (i.e. it admits the 1-variation) if and only if lim ε → ε R T | f ( s + ε ) − f ( s ) | ds exists. In this case, the previous limit equals the total variation of f . An objective was to producea more efficient stochastic calculus tool, able to go beyond the case of semimartingales. Given twoprocesses X and Y , their covariation [ X, Y ] ( t ) is the limit in probability, when ε goes to zero, of[ X, Y ] ε ( t ) = 1 ε Z t (cid:0) X ( s + ε ) − X ( s ) (cid:1)(cid:0) Y ( s + ε ) − Y ( s ) (cid:1) ds ; t ≥ . (3)2he limit is again denoted by [ X, Y ] ( t ) and this notion coincides with the classical covariationwhen X, Y are continuous semimartingales. The processes X such that [ X, X ] exists are calledfinite quadratic variation processes; their analysis and their applications were performed in [10, 26].The notion of covariation was also extended in order to involve more processes. In [9] theauthors consider the n -covariation [ X , X , · · · , X n ] of n processes X , . . . , X n , as in formula (3),but with a product of n increments rather than just two. For n = 4, for X being an fBm withso-called Hurst parameter H = 1 /
4, the paper [13] calculates the 4-covariation [ g ( X ) , X, X, X ]where g is, for instance, a bounded continuous function. If X = X = X = X is a singlestochastic process, we denote [ X ; 3] := [ X, X, X ], which is called the cubic variation , and is one ofthe main topics of investigation in our article. Note that this variation involves the signed cubes( X ( s + ε ) − X ( s )) , which has the same sign as the increment X ( s + ε ) − X ( s ), unlike the case ofquadratic or 2-variation, or of the so-called strong cubic variation, where absolute values are usedinside the cube function. Consider the case where X is a fractional Brownian motion B H withHurst parameter H ∈ (0 , H > /
6, in the introduction of [14] is established thatthe cubic variation [ X,
3] equals zero, while in [14, Theorem 4.1 part 2(c)] it is shown that [ X, H < /
6. In the limiting case of H = 1 /
6, [14, Theorem 4.1 part 2(b)] showsthat the regularization approximation [ X, ε ( t ) converges in law to a normal for every t >
0. Thisphenomenon is confirmed in the related study of finite-difference approximating sequence of [ X, X,
3] ( t ) converges inlaw to a Gaussian variable; this was noted in [23, Theorem 10], where the authors prove thatmore is true: considered as a process depending on the upper endpoint of the time interval, theapproximation converges in law to κW where W is an independent Brownian motion, and κ is auniversal constant given by κ = 34 X r ∈ Z ( | r + 1 | + | r − | − | r | ) . Beyond a basic interest in the variations of non-semimartingale stochastic processes, the signif-icance of the cubic variation lies in its ability to guarantee the existence of (generalized symmetric)Stratonovich integrals, and their associated Itˆo formula, for highly irregular processes, notablyfBm with
H > / /
6; compare withthe near 1 / f : R → R is a function of class C and X has a strong cubic variation, then the following Itˆotype formula holds: f ( X t ) = f ( X ) + Z t f ′ ( X s ) d ◦ X − Z t f ′′′ ( X s ) d [ X,
3] ( s ) , (4)and the stochastic integral in the right-hand side is the symmetric-Stratonovich integral introducedfor instance in [28], while the other is a Lebesgue-Stieltjes integral. The problem is that until now3o examples are known of processes X which have a cubic variation [ X,
3] which exists but doesnot vanish. In [21], an analogous formula to (4) is obtained for the case X = B H with H = 1 / X,
3] with theterm κW , W being the independent Wiener process identified in [23], so that R t f ′′′ ( X s ) d [ X,
3] ( s )is merely defined in law as a conditionally Wiener integral. Following the regularization methodology of [27] or [28], the cubic variation of a process X , denotedby [ X,
3] ( t ), was defined similarly to [9] as the limit in probability or in the mean square, as ε → X, ε ( t ) := ε − Z t ( X ( s + ε ) − X ( s )) ds. This was already mentioned above. This [ X,
3] will be null for a deterministic function X as longas it is α -H¨older-continuous with α > /
3. But the main physical reason for being interested inthis cubic variation for random processes is that, because the cube function is symmetric, if theprocess X itself has some probabilistic symmetry as well (such as the Gaussian property and thestationarity of increments), then we can expect [ X,
3] to be 0 for much more irregular processesthan those which are almost-surely α -H¨older-continuous with α > /
3. As mentioned above, [9]proves that fBm has zero cubic variation as soon as
H > /
6, in spite of the fact that fBm is only α -H¨older-continuous almost surely for all α < H . This doubling improvement over the deterministicsituation is due exclusively to the random symmetries of fBm, as they combine with the fact thatthe cube function is odd. Typically for other, non-symmetric types of variations, H needs to belarger to guarantee existence of the variation, let alone nullity; for instance, when X is fBm, itsstrong cubic variation, defined as the limit in probability of ε − R t | X ( s + ε ) − X ( s ) | ds , exists for H ≥ / X is fBm with H = 1 /
6. We have already observedthat this threshold represents a critical value in terms of existence of cubic variation, for fBm:we mentioned that whether in the sense of regularization or of finite-difference, the approximatingsequences of [ X,
3] ( t ) converge in law to Gaussian laws. On the other hand, these normal con-vergences contrast with one further point in the study of variations for fBm: in our article, weshow as a preliminary result (Proposition 2 herein), that [ X, ε does not converge in probabilityfor H = 1 /
6. The non-convergence of [ X, ε in probability for H < / α -H¨older-continuous paths, and to what extent the threshold α > / x
7→ | x | m sgn( x ) with arbitrary integer or non-integer m > m th variation”, defined (when it exists in the mean-square sense) by[ X, m ] ( t ) := lim ε → ε − Z t | X ( s + ε ) − X ( s ) | m sgn ( X ( s + ε ) − X ( s )) ds. (5)The qualifier “odd” above, when applied to m = 3, can easily yield the term “odd cubic variation”,which has historically been called simply “cubic variation” as we noted before, as opposed to the4strong cubic variation” which involves absolute values; therefore in this article, we will systemat-ically use the qualifier “odd” for all higher order m th variations based on odd functions, but willtypically omit it for the cubic variation. This article provides answers to some of the above questions, both in Gaussian and non-Gaussiansettings, and we hope that it will stimulate work on resolving some of the remaining open problems.Specifically, we consider the process X defined on [0 , T ] by X ( t ) = Z T G ( t, s ) dM ( s ) (6)where M is a square-integrable martingale on [0 , T ], and G is a non-random measurable functionon [0 , T ] , which is square-integrable in s with respect to d [ M ] s for every fixed t . In other words, X is defined using a Volterra representation with respect to a square-integrable martingale. Thequadratic variations of these martingale-based convolutions was studied in [8].What we call the Gaussian case is that in which M is the standard Wiener process (Brownianmotion) W . The itemized list below is a summary of our results. Here, for the reader’s convenience,we have not spelled out the technical conditions which are needed for some of our results, indicatinginstead references to the precise theorem statements in the body of this article. Some conditionsbecome more restrictive as one move from simple Gaussian cases to non-Gaussian cases. Yet wecover much wider classes of processes than has been done in the past. The summary below alsoprovides indications of how wide a scope we reach, and has references to examples in the main bodyof the paper.One condition which appears in all cases, is essentially equivalent to requiring that all processes X that we consider are not more regular than standard Brownian motion, i.e. are not 1 / δ ( s, t ) := E h ( X ( t ) − X ( s )) i . This condition is not a restriction on the range of pathregularity, since the main interest of our results occurs around the H¨older exponent 1 /
6, or moregenerally the exponent 1 / (2 m ) for any m >
1: the processes with zero odd m th variation appearas those which are better than 1 / (2 m )-H¨older-continuous in the L (Ω)-sense. Processes which arebetter than 1 / m ≥
2, let[
X, m ] ε ( t ) := 1 ε Z t ds | X ( s + ε ) − X ( s ) | m sgn ( X ( s + ε ) − X ( s ))where sgn ( x ) is the sign function x/ | x | . The limit in probability of [ X, m ] ε ( t ) as ε → m th variation” of X at time t , denotd by [ X, m ] ( t ). Except for some results in Section 5,the results in this paper are stated without loss of generality for a fixed value of t ≤ T , and wetypically take t = T ; we occasionally drop the dependence on T , writing only [ X, m ] ε and [ X, m ]. • [ Homogeneous Gaussian case, odd powers:
Theorem 6 on page 13]. When X is Gaussianwith homogeneous increments (meaning δ ( s, t ) depends only on | t − s | ), for any odd integer m ≥ X has zero odd m th variation if and only if δ ( r ) = o (cid:0) r / (2 m ) (cid:1) for r near 0.5 This theorem does not require any assumptions beyond δ being increasing and concave. • [ Homogeneous Gaussian case, arbitrary real powers:
Theorem 10 on page 22]. Thesufficient condition of the result above holds for any integer m > , and for any real non-integer > – This theorem extends the previous result to even powers without requiring any additionalassumptions, and to all real powers under a technical regularity assumption on thecovariance which places no regularity restrictions on the paths of X (see Remark 11). • [ Non-homogeneous Gaussian case:
Theorem 8 on page 19]. When X is Gaussian withnon-homogeneous increments, for any odd integer m ≥
3, if δ ( s, s + r ) = o (cid:0) r / (2 m ) (cid:1) for r near 0 uniformly in s , and under a technical condition, X has zero odd m th variation. – The technical condition is a non-explosion assumption on the mixed partial derivativeof δ near the diagonal. It places no regularity restriction on the paths of X . Thedescription on page 20 shows that the condition is satisfied for the so-called Riemann-Liouville version of fBm, and for a wide class of Volterra-convolution-type Gaussianprocesses with inhomogeneous increments. • [ Non-Gaussian martingale case:
Theorem 12 on page 26]. Let m ≥ X is non-Gaussian as in (6), based on a martingale M whose quadratic variationprocess has a derivative with 2 m moments (the actual condition on M in the theorem isweaker), let Γ ( t ) = ( E [( d [ M ] /dt ) m ]) / (2 m ) and consider the Gaussian process Z ( t ) = Z T Γ ( s ) G ( t, s ) dW ( s ) . Under a technical integrability condition on planar increments of Γ G near the diagonal, if Z satisfies the conditions of Theorem 6 or Theorem 8, then X has zero odd m th variation. – Proposition 13 on page 30 provides examples of wide classes of martingales and kernelsfor which the assumptions of Theorem 12 are satisfied. Details on how to construct theseexamples, and how to evaluate their regularity properties, are given on page 4. – A key consequence of Proposition 13 and Theorem 12 is that this paper’s results extendfrom the Gaussian case to highly non-Gaussian situations, insofar as, for m an oddinteger, it is easy to construct a variety of martingales M with no more than m moments,which are comparable to their Gaussian analogues in terms of path regularity, and forwhich the corresponding X in (7) has null odd m th variation. This is explained on page4. – It is important to note that while the base process M used here is a martingale, theprocess X in (7) whose variation we study is as far from being a martingale as fBm is. • [ Itˆo formula:
Theorem 14 on page 35, and its corollary]. When m ≥ X is a Gaussian process with non-homogeneous increments such that δ ( s, s + r ) = o (cid:0) r / (2 m ) (cid:1) s , under some additional technical conditions, for every bounded measurablefunction g on R , lim ε → ε E "(cid:18)Z T du ( X u + ε − X u ) m g (cid:18) X u + ε + X u (cid:19)(cid:19) = 0 . If m = 3, by results in [14], Theorem 14 implies that for any f ∈ C ( R ) and t ∈ [0 , T ], theItˆo formula f ( X t ) = f ( X ) + R t f ′ ( X u ) d ◦ X u holds, where the integral is in the symmetric(generalized Stratonovich) sense. This formula is in Corollary 15 on page 36. – The scope of the technical conditions needed for the theorem and its corollary is dis-cussed immediately after the corollary. These conditions include similar monotonicityand concavity conditions as are used in the remainder of the article, plus some coer-civity conditions ensuring that the process X is not too far from having homogeneousincrements. The discussion after Corollary 15 establishes that the coercivity conditionsare satisfied in the homogeneous case. We finish this introduction with a description of recent work done by several other authors onproblems related to our preoccupations to some extent, in various directions. The authors ofthe paper [15] consider, as we do, stochastic processes which can be written as Volterra integralswith respect to martingales. In fact, they study the concept of “fractional martingale”, which isthe generalization of the so-called Riemann-Liouville fractional Brownian motion when the drivingnoise is a martingale. This is a special case of the processes we consider in Section 4, with K ( t, s ) =( t − s ) H − / . The authors’ motivation is to prove an analogue of the famous characterization ofBrownian motion as the only continuous square-integrable martingale with a quadratic variationequal to t . They provide similar necessary and sufficient conditions based on the 1 /H -variation fora process to be fractional Brownian motion. The paper [15] does not follow, however, the samemotivation as our work: for us, say in the case of m = 3, we study the threshold H > / q rather than the power- q function, they show that the thresholdvalue H = 1 / (2 q ) poses an interesting open problem, since above this threshold (but below H =1 − / (2 q )) one obtains Gaussian limits (these limits are conditionally Gaussian when weights arepresent, and can be represented as stochastic integrals with respect to an independent Brownianmotion), while below the threshold, degeneracy occurs. The behavior at the threshold was workedout for H = 1 / , q = 2 in [20], boasting an exotic correction term with an independent Brownianmotion, while the general open problem of Hermite variations with H = 1 / (2 q ) was settled in [19].More questions arise, for instance, with a similar result in [18] for H = 1 /
4, but this time withbidimensional fBm, in which two independent Brownian motions are needed to characterize theexotic correction term.The value H = 1 / • choosing to prove necessary and sufficient conditions for nullity of the cubic variation, aroundthe threshold regularity value H = 1 /
6, for Gaussian processes with homogeneous increments(this is a wider class than previously considered, showing in particular that self-similarity isnot related to the question of nullity of the cubic variation); • studying the nullity threshold for higher order “odd” power functions, with possibly non-integer order, showing that this property relies only on the symmetry of Gaussian processeswith homogeneous increments and on the symmetrization of the power functions; • showing that our method is able to consider processes that are far from Gaussian and stillyield sharp sufficient conditions for nullity of odd power variations, since our base noise maybe a generic martingale with only a few moments.The article has the following structure. Section 2 contains some formal definitions and notations.The basic theorems in the Gaussian case are in Section 3, where the homogeneous case, non-homogeneous case, and case of non-integer m are separated in three subsections. The use ofnon-Gaussian martingales is treated in Section 4. Section 5 presents the Itˆo formula. We recall our process X defined for all t ∈ [0 , T ] by X ( t ) = Z T G ( t, s ) dM ( s ) (7)where M is a square-integrable martingale on [0 , T ], and G is a non-random measurable functionon [0 , T ] , which is square-integrable in s with respect to d [ M ] s for every fixed t . For any realnumber m ≥
2, let the odd ε - m -th variation of X be defined by[ X, m ] ε ( T ) := 1 ε Z T ds | X ( s + ε ) − X ( s ) | m sgn ( X ( s + ε ) − X ( s )) . (8)The odd variation is different from the absolute (or strong) variation because of the presence of thesign function, making the function | x | m sgn ( x ) an odd function. In the sequel, in order to lightenthe notation, we will write ( x ) m for | x | m sgn ( x ). We say that X has zero odd m -th variation (inthe mean-squared sense) if the limit lim ε → [ X, m ] ε ( T ) = 0 (9)holds in L (Ω).The canonical metric δ of a stochastic process X is defined as the pseudo-metric on [0 , T ] givenby δ ( s, t ) = E h ( X ( t ) − X ( s )) i . covariance function of X is defined by Q ( s, t ) = E [ X ( t ) X ( s )] . The special case of a centered Gaussian process is of primary importance; then the process’s entiredistribution is characterized by Q , or alternately by δ and the variances var ( X ( t )) = Q ( t, t ), sincewe have Q ( s, t ) = (cid:0) Q ( s, s ) + Q ( t, t ) − δ ( s, t ) (cid:1) . We say that δ has homogeneous increments ifthere exists a function on [0 , T ] which we also denote by δ such that δ ( s, t ) = δ ( | t − s | ) . Below, we will refer to this situation as the homogeneous case . This is in contrast to usual usage ofthis appellation, which is stronger, since for example in the Gaussian case, it refers to the fact that Q ( s, t ) depends only on the difference s − t ; this would not apply to, say, standard or fractionalBrownian motion, while our definition does. In non-Gaussian settings, the usual way to interpretthe “homogeneous” property is to require that the processes X ( t + · ) and X ( · ) have the same law,which is typically much more restrictive than our definition.The goal of the next two sections is to define various general conditions under which a charac-terization of the limit in (9) being zero can be established. In particular, we aim to show that X has zero odd m -th variation for well-behaved M ’s and G ’s as soon as δ ( s, t ) = o (cid:16) | t − s | / (2 m ) (cid:17) , (10)and that this is a necessary condition in some cases. Although this is a mean-square condition, it canbe interpreted as a regularity (local) condition on X ; for example, when X is a Gaussian process withhomogeneous increments, this condition means precisely that almost surely, the uniform modulus ofcontinuity ω of X on any fixed closed interval, defined by ω ( r ) = sup {| X ( t ) − X ( s ) | : | t − s | < r } ,satisfies ω ( r ) = o (cid:16) r / log / (1 /r ) (cid:17) . The lecture notes [1], as well as the article [29], can beconsulted for this type of statement. We assume that X is centered Gaussian. Then we can write X as in formula (7) with M = W astandard Brownian motion. More importantly, beginning with the easiest case where m is an oddinteger, we can easily show the following. Lemma 1 If m is an odd integer ≥ , we have E h ([ X, m ] ε ( T )) i = 1 ε m − / X j =0 c j Z T Z t dtds Θ ε ( s, t ) m − j V ar [ X ( t + ε ) − X ( t )] j V ar [ X ( s + ε ) − X ( s )] j := ( m − / X j =0 J j where the c j ’s are constants depending only on j , and Θ ε ( s, t ) := E [( X ( t + ε ) − X ( t )) ( X ( s + ε ) − X ( s ))] . roof. The lemma is an easy consequence of the following formula, which can be found asLemma 5.2 in [14]: for any centered jointly Gaussian pair of r.v.’s (
Y, Z ), we have E [ Y m Z m ] = ( m − / X j =0 c j E [ Y Z ] m − j V ar [ X ] j V ar [ Y ] j . We may translate Θ ε ( s, t ) immediately in terms of Q , and then δ . We have:Θ ε ( s, t ) = Q ( t + ε, s + ε ) − Q ( t, s + ε ) − Q ( s, t + ε ) + Q ( s, t )= 12 (cid:2) − δ ( t + ε, s + ε ) + δ ( t, s + ε ) + δ ( s, t + ε ) − δ ( s, t ) (cid:3) (11)=: −
12 ∆ ( s,t );( s + ε,t + ε ) δ . (12)Thus Θ ε ( s, t ) appears as the opposite of the planar increment of the canonical metric over therectangle defined by its corners ( s, t ) and ( s + ε, t + ε ). Before finding sufficient and possibly necessary conditions for various Gaussian processes to havezero cubic (or m th) variation, we discuss the threshold case for the cubic variation of fBm. Recallthat when X is fBm with parameter H = 1 /
6, as mentioned in the Introduction, it is known from[14, Theorem 4.1 part (2)] that [ X, ε ( T ) converges in distribution to a non-degenerate normal law.However, there does not seem to be any place in the literature specifying whether the convergencemay be any stronger than in distribution. We address this issue here. Proposition 2
Let X be an fBm with Hurst parameter H = 1 / . Then X does not have a cubicvariation (in the mean-square sense), by which we mean that [ X, ε ( T ) has no limit in L (Ω) as ε → . In fact more is true: [ X, ε ( T ) has no limit in probability as ε → . In order to prove the proposition, we study the Wiener chaos representation and moments of[ X, ε ( T ) when X is fBm; X is given by (7) where W is Brownian motion and the kernel G iswell-known. Information on G and on the Wiener chaos generated by W can be found respectivelyin Chapters 5 and 1 of the textbook [22]. The covariance formula for an fBm X is R H ( s, t ) := E [ X ( t ) X ( s )] = 2 − (cid:16) s H + t H − | t − s | H (cid:17) . (13) Lemma 3
Fix ε > . Let ∆ X s := X ( s + ε ) − X ( s ) and ∆ G s ( u ) := G ( s + ε, u ) − G ( s, u ) . Then [ X, ε ( T ) = I + I =: 3 ε Z T ds Z T ∆ G s ( u ) dW ( u ) (cid:18)Z T | ∆ G s ( v ) | dv (cid:19) (14)+ 6 ε Z T dW ( s ) Z s dW ( s ) Z s dW ( s ) Z T " Y k =1 ∆ G s ( s k ) ds. (15)10 roof. The proof of this lemma is elementary. It follows from two uses of the multiplicationformula for Wiener integrals [22, Proposition 1.1.3], for instance. It can also be obtained directlyfrom Lemma 7 below, or using the Itˆo formula technique employed further below in finding anexpression for [
X, m ] ε ( T ) in Step 0 of the proof of Theorem 12 on page 26. All details are left tothe reader.The above lemma indicates the Wiener chaos decomposition of [ X, ε ( T ) into the term I ofline (14) which is in the first Wiener chaos (i.e. a Gaussian term), and the term I of line (15), inthe third Wiener chaos. The next two lemmas contain information on the behavior of each of thesetwo terms, as needed to prove Proposition 2. Lemma 4
The Gaussian term I converges to in L (Ω) as ε → . Lemma 5
The 3rd chaos term I is bounded in L (Ω) for all ε > , and does not converge in L (Ω) as ε → . Proof of Proposition 2.
We prove the proposition by contradiction. Assume [ X, ε ( T )converges in probability. For any p >
2, there exists c p depending only on p such that E [ |I | p ] ≤ c p (cid:16) E h |I | i(cid:17) p/ and E [ |I | p ] ≤ c p (cid:16) E h |I | i(cid:17) p/ ; this is a general fact about random variables infixed Wiener chaos, and can be proved directly using Lemma 3 and the Burkh¨older-Davis-Gundyinequalities. Therefore, since we have sup ε> ( E h |I | i + E h |I | i ) < ∞ by Lemmas 4 and 5, we alsoget sup ε> ( E [ |I + I | p ]) < ∞ for any p . Therefore, by uniform integrability, [ X, ε ( T ) = I + I converges in L (Ω). In L (Ω), the terms I and I are orthogonal. Therefore, I and I mustconverge in L (Ω) separately. This contradicts the non-convergence of I in L (Ω) obtained inLemma 5. Thus [ X, ε ( T ) does not converge in probability.To conclude this section, we only need to prove the above two lemmas. To improve readability,we write H instead of 1 / Proof of Lemma 4.
Reintroducing the notation X and Θ into the formula in Lemma 3, weget I = 3 ε Z T ds ( X ( s + ε ) − X ( s )) V ar ( X ( s + ε ) − X ( s ))and therefore, E h |I | i = 9 ε Z T Z t dtds Θ ε ( s, t ) V ar ( X ( t + ε ) − X ( t )) V ar ( X ( s + ε ) − X ( s ))We note here that E h |I | i coincides with what we called J in Lemma 1, but we will not use thisfact here. Instead, using the variances of fBm, E h |I | i = 92 ε − H Z T Z T dtds Cov [ X ( t + ε ) − X ( t ) ; X ( s + ε ) − X ( s )]= 92 ε − H V ar (cid:20)Z T ( X ( t + ε ) − X ( t )) dt (cid:21) = 92 ε − H V ar (cid:20)Z T + εT X ( t ) dt − Z ε X ( t ) dt (cid:21) . E h |I | i ≤ ε − H (cid:18)Z T + εT Z T + εT R H ( s, t ) dsdt + Z ε Z ε R H ( s, t ) dsdt (cid:19) ≤ ε − H (cid:16) ε ( T + ε ) H + ε H (cid:17) = O (cid:0) ε H (cid:1) , proving Lemma 4. Proof of Lemma 5.
By the proof of Lemma 3, and using the covariance formula (13) for fBm,we first get E h |I | i = 12 ε Z T Z t dtds (Θ ε ( s, t )) = 6 ε Z T Z t dtds (cid:16) | t − s + ε | H + | t − s − ε | H − | t − s | H (cid:17) . Again, this expression coincides with the term J from Lemma 1, but this will not be used in thisproof. We must take care of the absolute values, i.e. of whether ε is greater or less than t − s . Wedefine the “off-diagonal” portion of E h |I | i as ODI := 6 ε − Z T ε Z t − ε dtds (cid:16) | t − s + ε | H + | t − s − ε | H − | t − s | H (cid:17) . For s, t in the integration domain for the above integral, since ¯ t := t − s > ε , by two iteratedapplications of the Mean Value Theorem for the function x H on the intervals [¯ t − ε, ¯ t ] and [¯ t, ¯ t + ε ], | ¯ t + ε | H + | ¯ t − ε | H − t H = 2 H (2 H − ε ( ξ − ξ ) ξ H − for some ξ ∈ [¯ t − ε, ¯ t ] , ξ ∈ [¯ t, ¯ t + ε ], and ξ ∈ [ ξ , ξ ], and therefore |ODI | ≤ H | H − | ε − Z T ε Z t − ε (cid:16) ε · ε · ( t − s − ε ) H − (cid:17) dtds = 384 H | H − | ε Z T ε Z t − ε ( t − s − ε ) H − dtds = 384 H | H − | ε − H Z T ε h ε H − − ( t − ε ) H − i dt ≤ H | H − | − H T ε H − = 384 H | H − | − H T = 32243 T. where in the last line we substituted H = 1 /
6. Thus the “off-diagonal” term is bounded. Thediagonal part of I is DI := 6 ε − Z T Z tt − ε dtds (cid:16) | t − s + ε | H + | t − s − ε | H − | t − s | H (cid:17) = 6 ε − T Z ε d ¯ t (cid:16) | ¯ t + ε | H + | ¯ t − ε | H − | ¯ t | H (cid:17) = 6 ε − H T Z dr (cid:16) | r + 1 | H + | r − | H − | r | H (cid:17) dr = CT H = 1 /
6, yields that C is a universal constant. Thus the diagonal part DI of E [ |I | ] is constant. This proves that I is bounded in L (Ω), as announced. To concludethat it cannot converge in L (Ω), recall that from [14, Theorem 4.1 part (2)], [ X, ε ( T ) = I + I converges in distribution to a non-degenerate normal law. By Lemma 4, I converges to 0 in L (Ω). Therefore, I converges in distribution to a non-degenerate normal law; if it also convergedin L (Ω), since the 3rd Wiener chaos is closed in L (Ω), the limit would have to be in that samechaos, and thus would not have a non-degenerate normal law. This concludes the proof of Lemma5. We now study the homogeneous case in detail. We are ready to prove a necessary and sufficientcondition for having a zero m -th variation when m is an odd integer. Theorem 6
Let m > be an odd integer. Let X be a centered Gaussian process on [0 , T ] withhomogeneous increments; its canonical metric is δ ( s, t ) := E h ( X ( t ) − X ( s )) i = δ ( | t − s | ) where the univariate function δ is assumed to be increasing and concave on [0 , T ] . Then X haszero m th variation if and only if δ ( r ) = o (cid:0) r / (2 m ) (cid:1) . Proof.
Step 0: setup.
We denote by dδ the derivative, in the sense of measures, of δ ; weknow that dδ is a positive bounded measure on [0 , T ]. Using homogeneity, we also get V ar [ X ( t + ε ) − X ( t )] = δ ( ε ) . Using the notation in Lemma 1, we get J j = ε − δ j ( ε ) c j Z T dt Z t ds Θ ε ( s, t ) m − j . Step 1: diagonal.
Let us deal first with the diagonal term. We define the ε -diagonal D ε := { ≤ t − ε < s < t ≤ T } . Trivially using Cauchy -Schwarz’s inequality, we have | Θ ε ( s, t ) | ≤ p V ar [ X ( t + ε ) − X ( t )] V ar [ X ( s + ε ) − X ( s )] = δ ( ε ) . Hence, according to Lemma 1, the diagonal portion P ( m − / j =0 J j,D ε of E h ([ X, m ] ε ( T )) i can bebounded above, in absolute value, as: ( m − / X j =0 J j,D ε := ( m − / X j =0 ε − δ j ( ε ) c j Z Tε dt Z tt − ε ds Θ ε ( s, t ) m − j . ≤ ε m − / X j =0 c j Z Tε dt Z tt − ε dsδ m ( ε ) ≤ c · ε − δ m ( ε )13here cst denotes a constant whose value may change in the remainder of the article’s proofs (hereit depends only on δ and m ). The hypothesis on δ implies that the above converges to 0 as ε tendsto 0. Step 2: small t term . The term for t ∈ [0 , ε ] and any s ∈ [0 , t ] can be dealt with similarly, and isof a smaller order than the one in Step 1. Specifically we have | J j,S | := ε − δ j ( ε ) c j (cid:12)(cid:12)(cid:12)(cid:12)Z ε dt Z t ds Θ ε ( s, t ) m − j (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε − δ j ( ε ) c j δ m − j ) ( ε ) ε = c j δ m ( ε ) , which converges to 0 like o ( ε ). Step 3: off-diagonal.
Because of the homogeneity hypothesis, we can calculate from (11) that forany s, t in the ε -off diagonal set OD ε := { ≤ s < t − ε < t ≤ T } Θ ε ( s, t ) = (cid:0) δ ( t − s + ε ) − δ ( t − s ) (cid:1) − (cid:0) δ ( t − s ) − δ ( t − s − ε ) (cid:1) = Z t − s + εt − s dδ ( r ) − Z t − st − s − ε dδ ( r ) . (16)By the concavity hypothesis, we see that Θ ε ( s, t ) is negative in this off-diagonal set OD ε . Unfor-tunately, using the notation in Lemma 1, this negativity does not help us because the off-diagonalportion J j,OD of J j also involves the constant c j , which could itself be negative. Hence we need toestimate J j,OD more precisely.The constancy of the sign of Θ ε is still useful, because it enables our first operation in thisstep, which is to reduce the estimation of | J j,OD | to the case of j = ( m − /
2. Indeed, usingCauchy-Schwarz’s inequality and the fact that | Θ ε | = − Θ ε , we write | J j,OD | = ε − δ j ( ε ) | c j | Z Tε dt Z t − ε ds | Θ ε ( s, t ) | m − j = − ε − δ j ( ε ) | c j | Z Tε dt Z t − ε ds Θ ε ( s, t ) | Θ ε ( s, t ) | m − j − ≤ ε − δ j ( ε ) | c j | Z Tε dt Z t − ε ds ( − Θ ε ( s, t )) (cid:12)(cid:12) δ ( ε ) (cid:12)(cid:12) m − j − = ε − δ m − ( ε ) | c j | Z Tε dt Z t − ε ds ( − Θ ε ( s, t )) . It is now sufficient to show that the estimate for the case j = ( m − / Z Tε dt Z t − ε ds ( − Θ ε ( s, t )) ≤ cst · εδ (2 ε ) (17)We rewrite the planar increments of δ as in (16) to show what cancellations occur: with thenotation s ′ = t − s , − Θ ε ( s, t ) = − (cid:0) δ (cid:0) s ′ + ε (cid:1) − δ (cid:0) s ′ (cid:1)(cid:1) + (cid:0) δ (cid:0) s ′ (cid:1) − δ (cid:0) s ′ − ε (cid:1)(cid:1) = − Z s ′ + εs ′ dδ ( r ) + Z s ′ s ′ − ε dδ ( r ) . s to s ′ , and another to change [ s ′ − ε, s ′ ] to [ s ′ , s ′ + ε ], Z Tε dt Z t − ε ds ( − Θ ε ( s, t )) = Z Tε dt "Z tε ds ′ Z s ′ s ′ − ε dδ ( r ) − Z tε ds ′ Z s ′ + εs ′ dδ ( r ) = Z Tε dt "Z tε ds ′ Z s ′ s ′ − ε dδ ( r ) − Z tε ds ′ Z s ′ + εs ′ dδ ( r ) = Z Tε dt "Z t − ε ds ′′ Z s ′′ + εs ′′ dδ ( r ) − Z tε ds ′ Z s ′ + εs ′ dδ ( r ) = Z Tε dt "Z ε ds ′′ Z s ′′ + εs ′′ dδ ( r ) − Z tt − ε ds ′ Z s ′ + εs ′ dδ ( r ) (18)We may now invoke the positivity of dδ , to obtain Z Tε dt Z t − ε ds ( − Θ ε ( s, t )) ≤ Z Tε dt Z ε ds ′′ Z s ′′ + εs ′′ dδ ( r )= Z Tε dt Z ε ds ′′ (cid:0) δ (cid:0) s ′′ + ε (cid:1) − δ (cid:0) s ′′ (cid:1)(cid:1) ≤ Z Tε dt ε δ (2 ε ) ≤ T εδ (2 ε ) . This is precisely the claim in (17), which finishes the proof that for all j , | J j,OD | ≤ cst · ε − δ m (2 ε )for some constant cst . Combining this with the results of Steps 1 and 2, we obtain that E h ([ X, m ] ε ( T )) i ≤ cst · ε − δ m (2 ε )which implies the sufficient condition in the theorem. Step 4: necessary condition.
The proof of this part is more delicate than the above: it requires anexcellent control of the off-diagonal term, since it is negative and turns out to be of the same orderof magnitude as the diagonal term. We spell out the proof here for m = 3. The general case issimilar, and is left to the reader. Step 4.1: positive representation.
The next lemma uses the following chaos integral notation: forany n ∈ N , for g ∈ L ([0 , T ] n ), g symmetric in its n variables, then I n ( g ) is the multiple Wienerintegral of g over [0 , T ] n with respect to W . This lemma’s elementary proof is left to the reader. Lemma 7
Let f ∈ L ([0 , T ]) . Then I ( f ) = 3 | f | L ([0 ,T ]) I ( f ) + I ( f ⊗ f ⊗ f )Using this lemma, as well as definitions (7) and (8), recalling the notation ∆ G s ( u ) := G ( s + ε, u ) − G ( s, u ) already used in Lemma 3, and exploiting the fact that the covariance of two multiple Wienerintegrals of different orders is 0, we can write E h ([ X, ε ( T )) i = 1 ε Z T ds Z T dt E h ( X ( s + ε ) − X ( s )) ( X ( t + ε ) − X ( t )) i = 1 ε Z T ds Z T dt E h I (∆ G s ) I (∆ G t ) i = 9 ε Z T ds Z T dt E [ I (∆ G s ) I (∆ G t )] | ∆ G s | L ([0 ,T ]) | ∆ G t | L ([0 ,T ]) + 9 ε Z T ds Z T dt E h I (cid:16) (∆ G s ) ⊗ (cid:17) I (cid:16) (∆ G t ) ⊗ (cid:17)i . E [ I ( f ) I ( g )] = h f, g i L ([0 ,T ] ) , plus the fact that in our homogeneoussituation | ∆ G s | L ([0 ,T ]) = δ ( ε ) for any s . Hence the above equals9 δ ( ε ) ε Z T ds Z T dt h ∆ G s , ∆ G t i L ([0 ,T ]) + 9 ε Z T ds Z T dt D (∆ G s ) ⊗ , (∆ G t ) ⊗ E L ([0 ,T ] ) = 9 δ ( ε ) ε Z T ds Z T dt Z T du ∆ G s ( u ) ∆ G t ( u ) + 9 ε Z T ds Z T dt Z Z Z [0 ,T ] Y i =1 ( du i ∆ G s ( u i ) ∆ G t ( u i ))= 9 δ ( ε ) ε Z T du (cid:12)(cid:12)(cid:12)(cid:12)Z T ds ∆ G s ( u ) (cid:12)(cid:12)(cid:12)(cid:12) + 9 ε Z Z Z [0 ,T ] du du du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T ds Y i =1 (∆ G s ( u i )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Step 4.2: J as a lower bound . The above representation is extremely useful because it turns out,as one readily checks, that of the two summands in the last expression above, the first is what wecalled J and the second is J , and we can now see that both these terms are positive, which wasnot at all obvious before, since, as we recall, the off-diagonal contribution to either term is negativeby our concavity assumption. Nevertheless, we may now have a lower bound on the ε -variation byfinding a lower bound for the term J alone.Reverting to our method of separating diagonal and off-diagonal terms, and recalling by Step2 that we can restrict t ≥ ε , we have J = 9 δ ( ε ) ε Z Tε dt Z t ds Z T du ∆ G s ( u ) ∆ G t ( u )= 9 δ ( ε ) ε Z Tε dt Z t ds Θ ε ( s, t )= 9 δ ( ε ) ε Z Tε dt Z t ds (cid:0) δ ( t − s + ε ) − δ ( t − s ) − (cid:0) δ ( t − s ) − δ ( | t − s − ε | ) (cid:1)(cid:1) = J ,D + J ,OD where, performing the change of variables t − s sJ ,D := 9 δ ( ε ) ε Z Tε dt Z ε ds (cid:0) δ ( s + ε ) − δ ( s ) − (cid:0) δ ( s ) − δ ( ε − s ) (cid:1)(cid:1) J ,OD := 9 δ ( ε ) ε Z Tε dt Z tε ds (cid:0) δ ( s + ε ) − δ ( s ) − (cid:0) δ ( s ) − δ ( s − ε ) (cid:1)(cid:1) . Step 4.3: Upper bound on | J ,OD | . Using the calculations performed in Step 3 (note here that( m − / J ,OD = 9 δ ( ε ) ε Z Tε dt (cid:20)Z tt − ε ds Z s + εs dδ ( r ) − Z ε ds Z s + εs dδ ( r ) (cid:21) =: K + K . We can already see that K ≥ K ≤
0, so it’s only necessary to find an upper bound on | K | ; but in reality, the reader will easily check that | K | is of the order δ ( ε ), and we will see that16his is much smaller than either J ,D or | K | . Performing a Fubini on the variables s and r , theintegrand in K is calculated as Z ε ds Z s + εs dδ ( r ) = Z εr =0 dδ ( r ) Z rs =0 ds + Z εr = ε dδ ( r ) Z εs = r − ε ds = Z εr =0 r dδ ( r ) + Z εr = ε (2 ε − r ) dδ ( r )= (cid:2) rδ ( r ) (cid:3) ε − (cid:2) rδ ( r ) (cid:3) εε − Z ε δ ( r ) dr + Z εε δ ( r ) dr + 2 ε (cid:0) δ (2 ε ) − δ ( ε ) (cid:1) = − Z ε δ ( r ) dr + Z εε δ ( r ) dr. In particular, because | K | ≪ | K | and δ is increasing, we get | J ,OD | ≤ δ ( ε ) ε Z Tε dt (cid:18)Z εε δ ( r ) dr − Z ε δ ( r ) dr (cid:19) = 9 ( T − ε ) δ ( ε ) ε (cid:18)Z εε δ ( r ) dr − Z ε δ ( r ) dr (cid:19) . (19) Step 4.4: Lower bound on J ,D . Note first that Z ε ds (cid:0) δ ( s ) − δ ( ε − s ) (cid:1) = Z ε ds δ ( s ) − Z ε ds δ ( ε − s ) = 0 . Therefore J ,D = 9 δ ( ε ) ε Z Tε dt Z ε ds (cid:0) δ ( s + ε ) − δ ( s ) (cid:1) = 9 δ ( ε ) ε ( T − ε ) Z ε ds Z s + εs dδ ( r ) . We can also perform a Fubini on the integral in J ,D , obtaining Z ε ds Z s + εs dδ ( r ) = Z ε r dδ ( r ) + ε Z εε dδ ( r )= (cid:2) rδ ( r ) (cid:3) ε − Z ε δ ( r ) dr + ε (cid:0) δ (2 ε ) − δ ( ε ) (cid:1) = εδ (2 ε ) − Z ε δ ( r ) dr. In other words, J ,D = 9 δ ( ε ) ε ( T − ε ) (cid:18) εδ (2 ε ) − Z ε δ ( r ) dr (cid:19) . tep 4.5: conclusion. We may now compare J ,D and | J ,OD | : using the results of Steps 4.1 and4.2, J = J ,D − | J ,OD | ≥ δ ( ε ) ε ( T − ε ) (cid:18) εδ (2 ε ) − Z ε δ ( r ) dr (cid:19) − δ ( ε ) ε ( T − ε ) (cid:18)Z εε δ ( r ) dr − Z ε δ ( r ) dr (cid:19) = 9 δ ( ε ) ε ( T − ε ) Z εε (cid:0) δ (2 ε ) − δ ( r ) (cid:1) dr. When δ is in the H¨older scale δ ( r ) = r H , the above quantity is obviously commensurate with δ ( ε ) /ε , which implies the desired result, but in order to be sure we are treating all cases, we nowpresent a general proof which only relies on the fact that δ is increasing and concave.Below we use the notation (cid:0) δ (cid:1) ′ for the density of dδ , which exists a.e. since δ is concave.The mean value theorem and the concavity of δ then imply that for any r ∈ [ ε, ε ], δ (2 ε ) − δ ( r ) ≥ (2 ε − r ) inf [ ε, ε ] (cid:0) δ (cid:1) ′ = (2 ε − r ) (cid:0) δ (cid:1) ′ (2 ε ) . Thus we can write J ≥ T − ε ) ε − δ ( ε ) (cid:0) δ (cid:1) ′ (2 ε ) Z εε (2 ε − r ) dr = 9( T − ε ) ε − δ ( ε ) (cid:0) δ (cid:1) ′ (2 ε ) ε / ≥ cst · δ ( ε ) · (cid:0) δ (cid:1) ′ (2 ε ) . Since δ is concave, and δ (0) = 0, we have δ ( ε ) ≥ δ (2 ε ) /
2. Hence, with the notation f ( x ) = δ (2 x ), we have J ≥ cst · f ( ε ) f ′ ( ε ) = cst · (cid:0) f (cid:1) ′ ( ε ) . Therefore we have that lim ε → (cid:0) f (cid:1) ′ ( ε ) = 0. We prove this implies lim ε → ε − f ( ε ) = 0. Indeed,fix η >
0; then there exists ε η > ε ∈ (0 , ε η ], 0 ≤ (cid:0) f (cid:1) ′ ( ε ) ≤ η (we used thepositivity of (cid:0) δ (cid:1) ′ ). Hence, also using f (0) = 0, for any ε ∈ (0 , ε η ],0 ≤ f ( ε ) ε = 1 ε Z ε (cid:0) f (cid:1) ′ ( x ) dx ≤ ε Z ε ηdx = η. This proves that lim ε → ε − f ( ε ) = 0, which is equivalent to the announced necessary condition,and finishes the proof of the theorem. The concavity and homogeneity assumptions were used heavily above for the proof of the necessarycondition in Theorem 6. However, these assumptions can be considerably weakened while stillresulting in a sufficient condition. We now show that a weak uniformity condition on the variances,coupled with a natural bound on the second-derivative measure of δ , result in zero m -variationprocesses. 18 heorem 8 Let m > be an odd integer. Let X be a centered Gaussian process on [0 , T ] withcanonical metric δ ( s, t ) := E h ( X ( t ) − X ( s )) i . Define a univariate function on [0 , T ] , also denoted by δ , via δ ( r ) := sup s ∈ [0 ,T ] δ ( s, s + r ) , and assume that for r near , δ ( r ) = o (cid:16) r / m (cid:17) . (20) Assume that, in the sense of distributions, the derivative ∂δ / ( ∂s∂t ) is a finite signed σ finitemeasure µ on [0 , T ] − ∆ where ∆ is the diagonal { ( s, s ) | s ∈ [0 , T ] } . Denote the off-diagonalsimplex by OD = { ( s, t ) : 0 ≤ s ≤ t − ε ≤ T } ; assume µ satisfies, for some constant c and for all ε small enough, | µ | ( OD ) ≤ cε − ( m − /m , (21) where | µ | is the total variation measure of µ . Then X has zero m th variation. Proof.
Step 0: setup.
Recall that by Lemma 1, E h ([ X, m ] ε ( T )) i = 1 ε m − / X j =0 c j Z T Z t dtds Θ ε ( s, t ) m − j δ j ( s, s + ε ) δ j ( t, t + ε ) (22):= ( m − / X j =0 J j and now we express Θ ε ( s, t ) = µ ([ s, s + ε ] × [ t, t + ε )) = Z s + εs Z t + εt µ ( dudv ) . (23)We again separate the diagonal term from the off-diagonal term, although this time the diagonalis twice as wide: it is defined as { ( s, t ) : 0 ≤ t − ε ≤ s ≤ t } . Step 1: diagonal.
Using Cauchy-Schwarz’s inequality which implies | Θ ε ( s, t ) | ≤ δ ( s, s + ε ) δ ( t, t + ε ),and bounding each term δ ( s, s + ε ) by δ ( ε ), the diagonal portion of E h ([ X, m ] ε ( T )) i can bebounded above, in absolute value, by1 ε m − / X j =0 c j Z T ε dt Z tt − ε dsδ m ( ε ) = cst · ε − δ m ( ε ) . The hypothesis on the univariate δ implies that this converges to 0 as ε tends to 0. The case of t ≤ ε works equally easily. Step 2: off diagonal.
The off-diagonal contribution is the sum for j = 0 , · · · , ( m − / J j,OD = ε − c j Z T ε dt Z t − ε dsδ j ( s, s + ε ) δ j ( t, t + ε ) Θ ε ( s, t ) m − j (24)19s we will prove below, the dominant term turns out to be J ( m − / ,OD ; we deal with it now. Step 2.1: term J ( m − / ,OD . Denoting c = (cid:12)(cid:12) c ( m − / (cid:12)(cid:12) , we have (cid:12)(cid:12) J ( m − / ,OD (cid:12)(cid:12) ≤ cδ m − ( ε ) ε Z T ε dt Z t − ε ds | Θ ε ( s, t ) | . We estimate the integral, using the formula (23) and Fubini’s theorem: Z T ε dt Z t − ε ds | Θ ε ( s, t ) | = Z T ε dt Z t − ε ds (cid:12)(cid:12)(cid:12)(cid:12)Z s + εs Z t + εt µ ( dudv ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z T ε dt Z t − ε ds Z s + εs Z t + εt | µ | ( dudv )= Z T + εv =2 ε Z min( v,T ) − εu =0 | µ | ( dudv ) Z min( v,T ) t =max(2 ε,v − ε,u + ε ) Z min( u,t − ε ) s =max(0 ,u − ε ) ds dt ≤ Z T + εv =2 ε Z v − εu =0 | µ | ( dudv ) Z vt = v − ε Z us = u − ε ds dt = ε Z T + εv =2 ε Z v − εu =0 | µ | ( dudv )Hence we have J ( m − / ,OD ≤ cδ m − ( ε ) Z T + εv =2 ε Z v − εu =0 | µ | ( dudv ) ≤ cδ m − ( ε ) | µ | ( OD ) , which again converges to 0 by hypothesis as ε goes to 0. Step 2.2: other J j,OD terms . Let now j < ( m − /
2. Using Cauchy-Schwarz’s inequality for allbut one of the m − j factors Θ in the expression (24) for J j,OD , which is allowed because m − j ≥ δ , we have | J j,OD | ≤ δ j ( ε ) c j ε Z T ε dt Z t − ε ds | Θ ε ( s, t ) | m − j − | Θ ε ( s, t ) |≤ δ m − ( ε ) c j ε − Z T ε dt Z t − ε ds | Θ ε ( s, t ) | , which is the same term we estimated in Step 2.1. This finishes the proof of the theorem.A typical situation covered by the above theorem is that of the Riemann-Liouville fractionalBrownian motion. This is the process B H,RL defined by B H,RL ( t ) = R t ( t − s ) H − / dW ( s ). Itscanonical metric is not homogeneous, but we do have, when H ∈ (0 , / | t − s | H ≤ δ ( s, t ) ≤ | t − s | H , (25)which implies, incidentally, that B H,RL has the same regularity properties as fractional Brownianmotion, see [17] for a proof of these inequalities. To apply the theorem, we must choose
H > / (2 m ) for the condition on the variances. For the other condition, we calculate that µ ( dsdt ) =2 H (1 − H ) | t − s | H − dsdt , and therefore µ ( OD ) = | µ | ( OD ) = c H Z T Z tε s H − dsdt ≤ c H T ε H − . This quantity is bounded above by ε − /m as soon as H ≥ / (2 m ), of course, so the strictinequality is sufficient to apply the theorem and conclude that B H,RL then has zero m th variation.One can generalize this example to any Gaussian process with a Volterra-convolution kernel:let γ be a univariate increasing concave function, differentiable everywhere except possibly at 0,and define X ( t ) = Z t (cid:18) dγ dr (cid:19) / ( t − r ) dW ( r ) . (26)Then one can show (see [17]) that the canonical metric δ ( s, t ) of X is bounded above by 2 γ ( | t − s | ),so that we can use the univariate δ = 2 γ , and also δ ( s, t ) is bounded below by γ ( | t − s | ).Similar calculations to the above then easily show that X has zero m th variation as soon as δ ( r ) = o (cid:0) r / (2 m ) (cid:1) . Hence there are inhomogeneous processes that are more irregular than frac-tional Brownian for any H > / (2 m ) which still have zero m th variation: use for instance the X above with γ ( r ) = r / (2 m ) / log (1 /r ). When m ≥ m th odd variation, we use the convention(( x )) m = | x | m sgn ( x ), which is an odd function. The idea here is to use the Taylor expansion forthis function up to order [ m ], with a remainder of order [ m ] + 1; it can be expressed as the followingelementary lemma, whose proof is omitted. Lemma 9
Fix m > and two reals a and b such that | a | ≥ | b | . Let (cid:0) mk (cid:1) denote the formal binomialcoefficient m ( m − · · · ( m − k + 1) / ( k ( k − · · · and let (( x )) := | x | sgn ( x ) . Then, for all reals a, b ,1. if | b/a | < , (( a )) m (( a + b )) m = [ m ] − X k =0 (cid:18) mk (cid:19) sgn k ( a ) | a | m − k b k + | a | m f m − (cid:18) ba (cid:19) ,
2. and if | a/b | < a )) m (( a + b )) m = [ m ] X k =0 (cid:18) mk (cid:19) sgn k +1 ( a ) sgn k +1 ( b ) | a | m + k | b | m − k + ( ab ) m f m (cid:16) ab (cid:17) , where for all | x | < , | f m ( x ) | ≤ c m | x | [ m ]+1 where c m depends only on m . When m is an integer,the above formulas have null remainder terms f . m th varia-tion, we are able to prove that the sufficient condition of Theorem 6 still works. The result is thefollowing. Theorem 10
Let X be as in Theorem 6 ( X with homogeneous increments, with an increasing andconcave δ ). Let m be any real number > . Consider the condition (S) δ is twice differentiable, and for some c < , the function r (cid:12)(cid:12)(cid:12)(cid:0) δ (cid:1) ′′ ( r ) (cid:12)(cid:12)(cid:12) is decreasing andbounded above by cr − δ ( r ) .If m is not an integer, then X has zero odd m th variation as soon as δ ( r ) = o (cid:0) r − / (2 m ) (cid:1) andcondition (S) holds.If m is an integer, the same is true without needing condition (S). Remark 11
The technical Condition (S) is not a restriction on the range of regularity of X .Indeed, for all fBm’s that are more irregular than Brownian motion, we have (cid:12)(cid:12)(cid:12)(cid:0) δ (cid:1) ′′ ( r ) (cid:12)(cid:12)(cid:12) = 2 H (1 − H ) r − δ ( r ) which is indeed decreasing, and the constant c can be taken as / (this maximum isattained for H = 1 / ). For perturbations of the fBm scale, where δ ( r ) is of the order r H log β (1 /r ) for some β , Condition (S) is also typically satisfied. Beyond the H¨older scale, in cases where δ ( r ) is of the order log β (1 /r ) with β < − / , we will actually have a stronger upper-bound condition,of the type (cid:12)(cid:12)(cid:12)(cid:0) δ (cid:1) ′′ ( r ) (cid:12)(cid:12)(cid:12) = o (cid:0) r − δ ( r ) (cid:1) . That (cid:12)(cid:12)(cid:12)(cid:0) δ (cid:1) ′′ ( r ) (cid:12)(cid:12)(cid:12) be decreasing is typical of all Gaussianprocesses with homogeneous increments, even those which are more regular than Brownian motion. Proof of Theorem 10.
Step 0: setup.
Recall that with Y = X ( t + ε ) − X ( t ) and Z = X ( s + ε ) − X ( s ), we have E h ([ X, m ] ε ( T )) i = 2 ε Z T Z t dtds E [(( Y )) m (( Z )) m ] . Now introduce the shorthand notation σ = V ar [ Y ], τ = V ar [ Z ], and θ = E [ Y Z ] = Θ ε ( s, t ).Thus Y = σM where M is a standard normal r.v.. We can write the “linear regression” expansionof Z w.r.t. Y , using another standard normal r.v. N independent of M : Z = θσ M + ρτ N where ρ := − (cid:18) θστ (cid:19) ! / . Note that ρ is always well-defined and positive by Cauchy-Schwarz’s inequality. Therefore(( Y )) m (( Z )) m = σ m (( M )) m (cid:18)(cid:18) θσ M + ρτ N (cid:19)(cid:19) m = sgn ( θ ) σ m | θ | − m (( a )) m (( a + b )) m where a := θσ M and b := ρτ N. Y )) m (( Z )) m is the sum of the following four expressions: A := | ρτσNθM | < m ] − X k =0 (cid:18) mk (cid:19) sgn k +1 ( θ ) | θ | m − k σ k τ k ρ k | M | m − k N k sgn k ( M ) (27) A ′ := sgn ( θ ) | ρτσNθM | < | θ | m | M | m f m − (cid:18) ρτ σNθM (cid:19) (28) B := | ρτσNθM | > m ] X k =0 (cid:18) mk (cid:19) sgn k +2 ( θ ) | θ | k σ m − k τ m − k ρ m − k sgn k +1 ( N M ) | M | m + k | N | m − k (29) B ′ := sgn ( θ ) | ρτσNθM | > ( ρστ ) m ( M N ) m f m (cid:18) θMρτ σN (cid:19) (30) Step 1: cancellations in expectation calculation for A and B . In evaluating the ε - m th variation E h ([ X, m ] ε ( T )) i , terms in A and B containing odd powers of M and N will cancel, because ofthe symmetry of the normal law, of the fact that the indicator functions in the expressions for A and B above are even functions of M and of N , and of their independence. Hence we can performthe following calculations, where a m,k := E h | M | m − k | N | k i and b m,k := E h | M | m + k | N | m − k i arepositive constants depending only on m and k . Step 1.1: expectation of A . In this case, because of the term N k , the expectation of all the termsin (27) with k odd drop out. We can expand the term ρ k using the binomial formula, and thenperform a change of variables. We then have, with n = [ m ] − m ] is even, or n = [ m ] − m ] is odd, | E [ A ] | ≤ [ m ] − X k =0 k even (cid:18) mk (cid:19) | θ | m − k σ k τ k a m,k k/ X ℓ =0 (cid:18) kℓ (cid:19) ( − ℓ (cid:18) θστ (cid:19) ℓ = n/ X j =0 | θ | m − j ( στ ) j n X k =2 jk even (cid:18) mk (cid:19)(cid:18) kk/ − j (cid:19) ( − k/ − j a m,k ≤ n/ X j =0 | θ | m − j ( στ ) j c m,j where c m,j are positive constant depending only on m and j . In all cases, the portion of E h ([ X, m ] ε ( T )) i corresponding to A can be treated using the same method as in the proof of Theorem 6. Moreprecisely, after multiplying by ε − and integrating over s and t , as we should, each term in the lastsum above is of the same form as the term J j in the proof of Theorem 6, whose upper-estimationis the subject of Steps 1, 2, and 3 in that proof. The lowest power is attained when j = n/
2, i.e.,when [ m ] is even we have | θ | m − [ m ] , and when [ m ] is odd, we have | θ | m − [ m ] . In both cases, thepower is greater than 1. All other values of j correspond of course to higher powers of | θ | . Thismeans we can use Cauchy-Schwarz’s inequality to get the bound, valid for all j , | θ | m − j ( στ ) j = | θ | | θ | m − j − ( στ ) j ≤ | θ | ( στ ) m − , j = ( m − / m is was odd integer. Thus the portion of E h ([ X, m ] ε ( T )) i corresponding to A tends to 0 as ε →
0, as long as δ ( r ) = o (cid:0) r / (2 m ) (cid:1) . Step 1.2: expectation of B . This portion is dealt with similarly. Because of the term sgn k +1 ( N ),the expectation of all the terms in (29) with k even drop out. Contrary to the case of A , we do notneed to expand ρ m − k in a binomial series. Since k is now ≥
1, we simply use Cauchy-Schwarz’sinequality to write | θ | k ≤ | θ | ( στ ) k − . Of course, we also have ρ <
1. Hence | E [ B ] | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E | ρτσNθM | > m ] X k =1 k odd (cid:18) mk (cid:19) sgn k +2 ( θ ) | θ | k σ m − k τ m − k ρ m − k sgn k +1 ( M N ) | M | m + k | N | m − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | θ | ( στ ) m − m ] X k =1 k odd b m,k (cid:18) mk (cid:19) . (31)We see here in all cases that we are exactly in the same situation of the proof of Theorem 6 (again,power of | θ | is | θ | ). Thus the portion of E h ([ X, m ] ε ( T )) i corresponding to B converges to 0 assoon as δ ( r ) = o (cid:0) r / (2 m ) (cid:1) . Step 2. The error term A ′ . For A ′ given in (28), we immediately have (cid:12)(cid:12) E (cid:2) A ′ (cid:3)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) E (cid:20) | ρτσNθM | < | θ | m | M | m f m − (cid:18) ρτ σNθM (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c m | θ | m E " | ρτσNθM | < | M | m (cid:12)(cid:12)(cid:12)(cid:12) ρτ σNθM (cid:12)(cid:12)(cid:12)(cid:12) [ m ] = c m | θ | m − [ m ] ( ρτ σ ) [ m ] E h | ρτσNθM | < | M | m − [ m ] | N | [ m ] i ‘ . We see here that we cannot ignore the indicator function inside the expectation, because if we didwe would be left with | θ | to the power m − [ m ], which is less than 1, and therefore does not allowus to use the proof of Theorem 6.To estimate the expectation, let x = ρτσ | θ | . We can use H¨older’s inequality for a conjugate pair p, q with p very large, to write E h | ρτσNθM | < | M | m − [ m ] | N | [ m ] i ≤ P /q [ | xN | < | M | ] E /p h | M | mp − [ m ] p | N | [ m ] p i . The second factor in the right-hand side above is a constant c m,p depending only on m and p . For thefirst factor, we can use the following standard estimate for all y > R ∞ y e − z / dz ≤ cst y − e − y / .Therefore, P [ | xN | < | M | ] = 2 Z ∞ du √ π e − u / P [ | M | > xu ] ≤ r π Z /x du e − u / + r π Z ∞ /x du e − u / ux e − u x / ≤ p /π x + p /π Z ∞ /x du ux e − u x / = p /π (cid:18) x + 1 x Z ∞ dv v e − v / (cid:19) = cx where c is a universal constant. 24ow choose q so that m − [ m ] + 1 /q = 1, i.e. q = (1 − m + [ m ]) − , which exceeds 1 as long as m is not an integer. Then we get (cid:12)(cid:12) E (cid:2) A ′ (cid:3)(cid:12)(cid:12) ≤ c m | θ | m − [ m ] ( ρτ σ ) [ m ] c m,p (cid:18) c | θ | ρστ (cid:19) /q = c m c m,p c /q | θ | ( ρτ σ ) m − , and we are again back to the usual computations. The case of m integer is dealt with in Step 4below. Step 3. The error term B ′ . For B ′ in (30), we have (cid:12)(cid:12) E (cid:2) B ′ (cid:3)(cid:12)(cid:12) ≤ c m σ m τ m ρ m E " | M N | m (cid:12)(cid:12)(cid:12)(cid:12) θMρτ σN (cid:12)(cid:12)(cid:12)(cid:12) [ m ]+1 = c m ( ρστ ) m − [ m ] − | θ | m ] E h | M | m +[ m ]+1 | N | − m − [ m ] i . The expectation above is a constant c ′ m depending only on m as long as m is not an integer. Thecase of m integer is trivial since then we have B ′ = 0. Now we can use Cauchy-Schwarz’s inequalityto say that | θ | [ m ] ≤ ( στ ) [ m ] , yielding (cid:12)(cid:12) E (cid:2) B ′ (cid:3)(cid:12)(cid:12) ≤ c m c ′ m ρ m − [ m ] − ( στ ) m − | θ | . This is again identical to the terms we have already dealt with, but for the presence of the negativepower on ρ . We will handle this complication by showing that ρ can be bounded below by auniversal constant.First note that integration on the ε -diagonal can be handled by using the same argumentas in Steps 1 and 2 of the proof of Theorem 6. Thus we can assume that t ≥ s + 2 ε . Nowthat we are off the diagonal, note that using the mean value theorem on the expression for θ in (11), we can write that θ = ε (cid:0) δ (cid:1) ′′ ( ξ ) for some ξ in [ t − s − ε, t − s + ε ]. At this point,Condition (S) allows us to say first that (cid:12)(cid:12)(cid:12)(cid:0) δ (cid:1) ′′ ( r ) (cid:12)(cid:12)(cid:12) ≤ cr − δ ( r ). We now use the expression στ = δ ( ε ), and the fact that off the diagonal, ξ > ε , combined with the fact that (cid:12)(cid:12)(cid:12)(cid:0) δ (cid:1) ′′ (cid:12)(cid:12)(cid:12) isdecreasing according to Condition (S), to write | θ/ ( στ ) | ≤ ε (cid:12)(cid:12)(cid:12)(cid:0) δ (cid:1) ′′ ( ε ) (cid:12)(cid:12)(cid:12) δ − ( ε ) ≤ c . Recalling thedefinition of ρ := (cid:16) − θ ( στ ) − (cid:17) / , we have proved that ρ is bounded below uniformly (off the ε -diagonal) by the positive constant c ′′ := (cid:0) − c (cid:1) / . Hence, in the inequality for | E [ B ′ ] | above,the term ρ m − [ m ] − can be absorbed into the m -dependent constants. In other words, we haveproved the upper bound | E [ B ′ ] | ≤ c m c ′ m ( c ′′ ) m − [ m ] − ( στ ) m − | θ | , and we are back once again tothe situation solved in the proof of Theorem 6, proving the corresponding contribution of B ′ to E h ([ X, m ] ε ( T )) i converges to 0 as soon as δ ( r ) = o (cid:0) r / (2 m ) (cid:1) . Step 4. The case of m integer . Of course, we already proved the theorem in the case m odd. Nowassume m is an even integer. In this special case, we do not need to use a Taylor expansion, sincethe binomial formula has no remainder. Moreover, on the event (cid:12)(cid:12)(cid:12) ρτσNθM (cid:12)(cid:12)(cid:12) <
1, sgn ( a + b ) = sgn ( a ).Thus A ′ = 0 with the understanding that we must replace A by the full sum for k = 0 to m .25ecalculating this A we get A = σ m sgn ( θ ) sgn ( M ) | M | m | ρτσNθM | < m X k =0 (cid:18) mk (cid:19) M k N m − k (cid:20) θσ (cid:21) k [ ρτ ] m − k = sgn ( θ ) | ρτσNθM | < m X k =0 (cid:18) mk (cid:19) sgn k +1 ( M ) | M | m + k N m − k θ k ρ m − k σ m − k τ m − k . Here, when we take the expectation E , all terms vanish since we have odd functions of M for k even thanks to the term sgn k +1 ( M ), and odd functions of N for k odd thanks to the term N m − k .I.e. the term corresponding to A is entirely null when m is even. The term B ′ is null since we haveno error terms in the Taylor expansion. The estimation of the term B in Step 1.2 above appliedwhen m is an integer. The proof of the theorem is finished. Now assume that X is given by (7) and M is a square-integrable (non-Gaussian) martingale, m isan odd integer, and define a positive non-random measure µ for ¯ s = ( s , s , · · · , s m ) ∈ [0 , T ] m by µ ( d ¯ s ) = µ ( ds ds · · · ds m ) = E [ d [ M ] ( s ) d [ M ] ( s ) · · · d [ M ] ( s m )] , (32)where [ M ] is the quadratic variation process of M . We make the following assumption on µ . (A) The non-negative measure µ is absolutely continuous with respect to the Lebesgue measure d ¯ s on [0 , T ] m and K (¯ s ) := dµ/d ¯ s is bounded by a tensor-power function: 0 ≤ K ( s , s , · · · , s m ) ≤ Γ ( s ) Γ ( s ) · · · Γ ( s m ) for some non-negative function Γ on [0 , T ].A large class of processes satisfying (A) is the case where M ( t ) = R t H ( s ) dW ( s ) where H ∈ L ([0 , T ] × Ω) and W is a standard Wiener process, and we assume E (cid:2) H m ( t ) (cid:3) is finite for all t ∈ [0 , T ]. Indeed then by H¨older’s inequality, since we can take K (¯ s ) = E (cid:2) H ( s ) H ( s ) · · · H ( s m ) (cid:3) ,we see that Γ ( s ) = (cid:0) E (cid:2) H m ( t ) (cid:3)(cid:1) / (2 m ) works.We will show that the sufficient conditions for zero odd variation in the Gaussian cases generalizeto the case of condition (A), by associating X with a Gaussian process. We let˜ G ( t, s ) = Γ ( s ) G ( t, s )and Z ( t ) := Z T ˜ G ( t, s ) dW ( s ) . (33)We have the following. Theorem 12
Let m be an odd integer ≥ . Let X and Z be as defined in (7) and (33). Assume M satisfies condition (A) and Z is well-defined and satisfies the hypotheses of Theorem 6 or Theorem8 relative to a univariate function δ . Assume that for some constant c > , and every small ε > , Z Tt =2 ε dt Z t − εs =0 ds Z Tu =0 (cid:12)(cid:12)(cid:12) ∆ ˜ G t ( u ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∆ ˜ G s ( u ) (cid:12)(cid:12)(cid:12) du ≤ cεδ (2 ε ) , (34) where we use the notation ∆ ˜ G t ( u ) = ˜ G ( t + ε, u ) − ˜ G ( t, u ) . Then X has zero m th variation. roof. Step 0: setup.
We use an expansion for powers of martingales written explicitly at Corollary2.18 of [9]. For any integer k ∈ [0 , [ m/ km be the set of permutations σ of m − k de-fined as those for which the first k terms σ − (1) , σ − (2) , · · · , σ − ( k ) are chosen arbitrarily andthe next m − k terms are chosen arbitrarily among the remaining integers { , , · · · , m − k } \ (cid:8) σ − (1) , σ − (2) , · · · , σ − ( k ) (cid:9) . Let Y be a fixed square-integrable martingale. We define theprocess Y σ,ℓ (denoted in the above reference by σ ℓY ) by setting, for each σ ∈ Σ km and each ℓ = 1 , , · · · , m − k , Y σ,ℓ ( t ) = (cid:26) [ Y ] ( t ) if σ ( ℓ ) ∈ { , , · · · , k } Y ( t ) if σ ( ℓ ) ∈ { k + 1 , · · · , m − k } . From Corollary 2.18 of [9], we then have for all t ∈ [0 , T ]( Y t ) m = [ m/ X k =0 m !2 k X σ ∈ Σ km Z t Z u m − k · · · Z u dY σ, ( u ) dY σ, ( u ) · · · dY σ,m − k ( u m − k ) . We use this formula to evaluate[
X, m ] ε ( T ) = 1 ε Z T ds ( X ( s + ε ) − X ( s )) m by noting that the increment X ( s + ε ) − X ( s ) is the value at time T of the martingale Y t := R t ∆ G s ( u ) dM ( u ) where we set∆ G s ( u ) := G ( s + ε, u ) − G ( s, u ) . Hence( X ( s + ε ) − X ( s )) m = [ m/ X k =0 m !2 k X σ ∈ Σ km Z T Z u m − k · · · Z u d [ M ] (cid:0) u σ (1) (cid:1) (cid:12)(cid:12) ∆ G s (cid:0) u σ (1) (cid:1)(cid:12)(cid:12) · · · d [ M ] (cid:0) u σ ( k ) (cid:1) (cid:12)(cid:12) ∆ G s (cid:0) u σ ( k ) (cid:1)(cid:12)(cid:12) dM (cid:0) u σ ( k +1) (cid:1) ∆ G s (cid:0) u σ ( k +1) (cid:1) · · · dM (cid:0) u σ ( m − k ) (cid:1) ∆ G s (cid:0) u σ ( m − k ) (cid:1) . Therefore we can write[
X, m ] ε ( T )= 1 ε [ m/ X k =0 m !2 k X σ ∈ Σ km Z T Z u m − k · · · Z u d [ M ] (cid:0) u σ (1) (cid:1) · · · d [ M ] (cid:0) u σ ( k ) (cid:1) dM (cid:0) u σ ( k +1) (cid:1) · · · dM (cid:0) u σ ( m − k ) (cid:1)(cid:2) ∆ G · (cid:0) u σ ( k +1) (cid:1) ; · · · ; ∆ G · (cid:0) u σ ( m − k ) (cid:1) ; ∆ G · (cid:0) u σ (1) (cid:1) ; ∆ G · (cid:0) u σ (1) (cid:1) ; · · · ; ∆ G · (cid:0) u σ ( k ) (cid:1) ; ∆ G · (cid:0) u σ ( k ) (cid:1)(cid:3) , where we have used the notation[ f , f , · · · , f m ] := Z T f ( s ) f ( s ) · · · f m ( s ) ds. To calculate the expected square of the above, we will bound it above by the sum over k and σ of the expected square of each term. Writing squares of Lebesgue integrals as double integrals,27nd using Itˆo’s formula, each term’s expected square is thus, up to ( m, k )-dependent multiplicativeconstants, equal to the expression K = 1 ε Z Tu m − k =0 Z Tu ′ m − k =0 Z u m − k u m − k − =0 Z u m − k u ′ m − k − =0 · · · Z u u =0 Z u u ′ =0 E h d [ M ] ⊗ k (cid:0) u σ (1) , · · · , u σ ( k ) (cid:1) d [ M ] ⊗ k (cid:16) u ′ σ (1) , · · · , u ′ σ ( k ) (cid:17) d [ M ] ⊗ ( m − k ) (cid:0) u σ ( k +1) , · · · , u σ ( m − k ) (cid:1)i · (cid:2) ∆ G · (cid:0) u σ ( k +1) (cid:1) ; · · · ; ∆ G · (cid:0) u σ ( m − k ) (cid:1) ; ∆ G · (cid:0) u σ (1) (cid:1) ; ∆ G · (cid:0) u σ (1) (cid:1) ; · · · ; ∆ G · (cid:0) u σ ( k ) (cid:1) ; ∆ G · (cid:0) u σ ( k ) (cid:1)(cid:3) · h ∆ G · (cid:0) u σ ( k +1) (cid:1) ; · · · ; ∆ G · (cid:0) u σ ( m − k ) (cid:1) ; ∆ G · (cid:16) u ′ σ (1) (cid:17) ; ∆ G · (cid:16) u ′ σ (1) (cid:17) ; · · · ; ∆ G · (cid:16) u ′ σ ( k ) (cid:17) ; ∆ G · (cid:16) u ′ σ ( k ) (cid:17)i , (35)modulo the fact that one may remove the integrals with respect to those u ′ j ’s that are not rep-resented among { u ′ σ (1) , · · · , u ′ σ ( k ) } . The theorem will now be proved if we can show that for all k ∈ { , , , · · · , [ m/ } and all σ ∈ Σ km , the above expression K = K m,k,σ tends to 0 as ε tends to0. A final note about notation. The bracket notation in the last two lines of the expression (35)above means that we have the product of two separate Riemann integrals over s ∈ [0 , T ]. Belowwe will denote these integrals as being with respect to s ∈ [0 , T ] and t ∈ [0 , T ]. Step 1: diagonal.
As in Steps 1 of the proofs of Theorems 6 and 8, we can use brutal applicationsof Cauchy-Schwarz’s inequality to deal with the portion of K m,k,σ in (35) where | s − t | ≤ ε . Thedetails are omitted. Step 2: term for k = 0. When k = 0, there is only one permutation σ = Id , and we have, usinghypothesis (A) K m, ,Id = 1 ε Z Tu m =0 Z u m u m − =0 · · · Z u u =0 E (cid:2) d [ M ] ⊗ m ( u , · · · , u m ) (cid:3) · [∆ G · ( u ) ; · · · ; ∆ G · ( u m )] ≤ ε Z Tu m − k =0 Z u m − k u m − k − =0 · · · Z u u =0 Γ ( u ) Γ ( u ) · · · Γ ( u m ) [∆ G · ( u ) ; · · · ; ∆ G · ( u m )] du du · · · du m = 1 ε Z Tu m − k =0 Z u m − k u m − k − =0 · · · Z u u =0 h ∆ ˜ G · ( u ) ; · · · ; ∆ ˜ G · ( u m ) i du du · · · du m . This is precisely the expression one gets for the term corresponding to k = 0 when M = W , i.e.when X is the Gaussian process Z with kernel ˜ G . Hence our hypotheses from the previous twotheorems guarantee that this expression tends to 0.28 tep 3: term for k = 1. Again, in this case, there is only one possible permutation, σ = Id , andwe thus have, using hypothesis (A), K m, ,Id = 1 ε Z Tu m − =0 Z u m − u m − =0 · · · Z u u =0 Z u u ′ =0 E h d [ M ] ( u ) d [ M ] (cid:0) u ′ (cid:1) d [ M ] ⊗ ( m − ( u , · · · , u m − ) i · [∆ G · ( u ) ; · · · ; ∆ G · ( u m − ) ; ∆ G · ( u ) ; ∆ G · ( u )] · (cid:2) ∆ G · ( u ) ; · · · ; ∆ G · ( u m − ) ; ∆ G · (cid:0) u ′ (cid:1) ; ∆ G · (cid:0) u ′ (cid:1)(cid:3) ≤ ε Z Tu m − =0 Z u m − u m − =0 · · · Z u u =0 Z u u ′ =0 du du ′ du · · · du m − Γ ( u ) Γ (cid:0) u ′ (cid:1) Γ ( u ) · · · Γ ( u m ) · [ | ∆ G | · ( u ) ; · · · ; | ∆ G | · ( u m − ) ; | ∆ G | · ( u ) ; | ∆ G | · ( u )] · (cid:2) | ∆ G | · ( u ) ; · · · ; | ∆ G | · ( u m − ) ; | ∆ G | · (cid:0) u ′ (cid:1) ; | ∆ G | · (cid:0) u ′ (cid:1)(cid:3) = 1 ε Z Tu m − =0 Z u m − u m − =0 · · · Z u u =0 Z u u ′ =0 du du ′ du · · · du m − h(cid:12)(cid:12)(cid:12) ∆ ˜ G (cid:12)(cid:12)(cid:12) · ( u ) ; · · · ; (cid:12)(cid:12)(cid:12) ∆ ˜ G (cid:12)(cid:12)(cid:12) · ( u m − ) ; (cid:12)(cid:12)(cid:12) ∆ ˜ G (cid:12)(cid:12)(cid:12) · ( u ) ; (cid:12)(cid:12)(cid:12) ∆ ˜ G (cid:12)(cid:12)(cid:12) · ( u ) i · h(cid:12)(cid:12)(cid:12) ∆ ˜ G (cid:12)(cid:12)(cid:12) · ( u ) ; · · · ; (cid:12)(cid:12)(cid:12) ∆ ˜ G (cid:12)(cid:12)(cid:12) · ( u m − ) ; (cid:12)(cid:12)(cid:12) ∆ ˜ G (cid:12)(cid:12)(cid:12) · (cid:0) u ′ (cid:1) ; (cid:12)(cid:12)(cid:12) ∆ ˜ G (cid:12)(cid:12)(cid:12) · (cid:0) u ′ (cid:1)i Note now that the product of two bracket operators [ · · · ] [ · · · ] means we integrate over 0 ≤ s ≤ t − ε and 2 ε ≤ t ≤ T , and get an additional factor of 2, since the diagonal term was dealt with in Step 1.In order to exploit the additional hypothesis (34) in our theorem, our first move is to use Fubiniby bringing the integrals over u all the way inside. We get K m, ,Id ≤ ε Z Tu m − =0 Z u m − u m − =0 · · · Z u u =0 du · · · du m − Z Tt =2 ε Z t − εs =0 ds dt (cid:12)(cid:12)(cid:12) ∆ ˜ G s ( u ) (cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12) ∆ ˜ G s ( u m − ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∆ ˜ G t ( u ) (cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12) ∆ ˜ G t ( u m − ) (cid:12)(cid:12)(cid:12)Z u u =0 Z u u ′ =0 du du ′ (cid:16) ∆ ˜ G s ( u ) (cid:17) (cid:16) ∆ ˜ G t (cid:0) u ′ (cid:1)(cid:17) . The term in the last line above is trivially bounded above by Z Tu =0 Z Tu ′ =0 du du ′ (cid:16) ∆ ˜ G s ( u ) (cid:17) (cid:16) ∆ ˜ G t (cid:0) u ′ (cid:1)(cid:17) precisely equal to V ar [ Z ( s + ε ) − Z ( s )] V ar [ Z ( t + ε ) − Z ( t )], which by hypothesis is boundedabove by δ ( ε ). Consequently, we get K m, ,Id ≤ δ ( ε ) ε Z Tu m − =0 Z u m − u m − =0 · · · Z u u =0 du · · · du m − Z Tt =2 ε Z t − εs =0 ds dt (cid:12)(cid:12)(cid:12) ∆ ˜ G s ( u ) (cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12) ∆ ˜ G s ( u m − ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∆ ˜ G t ( u ) (cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12) ∆ ˜ G t ( u m − ) (cid:12)(cid:12)(cid:12) .
29e get an upper bound by integrating all the u j ’s over their entire range [0 , T ]. I.e. we have, K m, ,Id ≤ δ ( ε ) ε Z Tt =2 ε dt Z t − εs =0 ds Z T Z T · · · Z T du · · · du m − (cid:12)(cid:12)(cid:12) ∆ ˜ G s ( u ) (cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12) ∆ ˜ G s ( u m − ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∆ ˜ G t ( u ) (cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12) ∆ ˜ G t ( u m − ) (cid:12)(cid:12)(cid:12) · Z Tu =0 (cid:12)(cid:12)(cid:12) ∆ ˜ G t ( u ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∆ ˜ G s ( u ) (cid:12)(cid:12)(cid:12) du = 2 δ ( ε ) ε Z Tt =2 ε dt Z t − εs =0 ds (cid:18)Z T du (cid:12)(cid:12)(cid:12) ∆ ˜ G s ( u ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∆ ˜ G t ( u ) (cid:12)(cid:12)(cid:12)(cid:19) m − · Z u u =0 (cid:12)(cid:12)(cid:12) ∆ ˜ G t ( u ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∆ ˜ G s ( u ) (cid:12)(cid:12)(cid:12) du .. Now we use a simple Cauchy-Schwarz inequality for the integral over u , but not for u . Recognizingthat R T (cid:12)(cid:12)(cid:12) ∆ ˜ G s ( u ) (cid:12)(cid:12)(cid:12) du is the variance V ar [ Z ( s + ε ) − Z ( s )] ≤ δ ( ε ), we have K m, ,Id ≤ δ ( ε ) ε Z Tt =2 ε dt Z t − εs =0 ds (cid:18)Z T du (cid:12)(cid:12)(cid:12) ∆ ˜ G s ( u ) (cid:12)(cid:12)(cid:12) (cid:19) m − · Z u u =0 (cid:12)(cid:12)(cid:12) ∆ ˜ G t ( u ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∆ ˜ G s ( u ) (cid:12)(cid:12)(cid:12) du . ≤ δ m − ( ε ) ε Z Tt =2 ε dt Z t − εs =0 ds Z Tu =0 (cid:12)(cid:12)(cid:12) ∆ ˜ G t ( u ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∆ ˜ G s ( u ) (cid:12)(cid:12)(cid:12) du . Condition (34) implies immediately K m, ,Id ≤ δ m (2 ε ) ε − which tends to 0 with ε by hypothesis. Step 4: k ≥
2. This step proceeds using the same technique as Step 3. Fix k ≥
2. Now for eachgiven permutation σ , there are k pairs of parameters of the type ( u, u ′ ). Each of these contributesprecisely a term δ ( ε ), as in the previous step, i.e. δ k ( ε ) altogether. In other words, for every σ ∈ Σ km , and deleting the diagonal term, we have K m,k,σ ≤ δ k ( ε ) ε Z Tt =2 ε dt Z t − εs =0 ds Z T Z u m − k · · · Z u k +2 du k +1 · · · du m − k (cid:20)Z T ds (cid:12)(cid:12)(cid:12) ∆ ˜ G s ( u k +1 ) (cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12) ∆ ˜ G s ( u m − k ) (cid:12)(cid:12)(cid:12)(cid:21) . Since k ≤ ( m − /
2, there is at least one integral, the one in u k +1 , above. We treat all theremaining integrals, if any, over u k +2 , · · · , u m − k with Cauchy-Schwarz’s inequality as in Step 3,yielding a contribution δ m − k − ( ε ). The remaining integral over u k +1 yields, by Condition (34),a contribution of δ (2 ε ) ε . Hence the contribution of K m,k,σ is again δ m (2 ε ) ε − , which tends to0 with ε by hypothesis, concluding the proof of the Theorem.We state and prove the next proposition, in order to illustrate the range of applicability ofTheorem 12. It provides a class of martingale-based processes X which can be associated to non-homogeneous Gaussian processes Z satisfying the assumptions of Theorem 8 and the additionalassumption (34). Proposition 13
Let X be defined by (7) via the kernel G and the martingale M . Assume m isan odd integer ≥ and condition (A) holds. Assume that ˜ G ( t, s ) := Γ ( s ) G ( t, s ) can be boundedabove as follows: for all s, t , ˜ G ( t, s ) = s ≤ t g ( t, s ) = s ≤ t | t − s | / (2 m ) − / f ( t, s )30 n which the bivariate function f ( t, s ) is positive and bounded as | f ( t, s ) | ≤ f ( | t − s | ) where the univariate function f ( r ) is increasing, and concave on R + , with lim r → f ( r ) = 0 , andwhere g has a second mixed derivative such that (cid:12)(cid:12)(cid:12)(cid:12) ∂g∂t ( t, s ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂g∂s ( t, s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c | t − s | / (2 m ) − / ; (cid:12)(cid:12)(cid:12)(cid:12) ∂ g∂s∂t ( t, s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c | t − s | / (2 m ) − / . Also assume that g is decreasing in t and the bivariate f is increasing in t . Then X has zero m -variation. The presence of the indicator function s ≤ t in the expression for ˜ G above is typical of mostmodels, since it coincides with asking that Z be adapted to the filtrations of W , which is equivalentto X being adapted to the filtration of M . In the case of irregular processes, which is the focus ofthis paper, the presence of the indicator function makes ˜ G non-monotone in both s and t , whichcreates technical difficulties. Examples of non-adapted irregular processes are easier to treat, since itis possible to require that ˜ G be monotone. We do not consider such non-adapted processes further.Specific examples of adapted processes which fall in the class defined in the above proposition aregiven below, after the proposition’s proof. Proof of Proposition 13.
Below the value 1 / (2 m ) − / α . We now show thatwe can apply Theorem 8 directly to the Gaussian process Z given in (33), which, by Theorem 12,is sufficient, together with Condition (34), to obtain our desired conclusion. Note the assumptionabout ˜ G implies that s ˜ G ( t, s ) is square-integrable, and therefore Z is well-defined. We willprove Condition (20) holds in Step 1; Step 2 will show Condition (21) holds; Condition (34) willbe established in Step 3. Step 1. Variance calculation.
We need only to show ˜ δ ( s, s + ε ) = o (cid:0) ε /m (cid:1) uniformly in s . Wehave, for given s and t = s + ε ˜ δ ( s, s + ε ) = Z t (cid:12)(cid:12)(cid:12) ˜ G ( t, r ) − ˜ G ( s, r ) (cid:12)(cid:12)(cid:12) dr = Z s | ( s + ε − r ) α f ( s + ε, r ) − ( s − r ) α f ( s, r ) | dr + Z s + εs | s + ε − r | α f ( s + ε, r ) dr (36)=: A + B. Since f ( s + ε, r ) ≤ f ( s + ε − r ) and the univariate f increases, in B we can bound this lastquantity by f ( ε ), and we get B ≤ f ( ε ) Z ε r α dr = 3 f ( ε ) ε α +1 = o (cid:16) ε /m (cid:17) . A is slightly more delicate to estimate. By the fact that f is increasing and g isdecreasing in t , A ≤ Z s f ( s + ε, r ) | ( s + ε − r ) α − ( s − r ) α | dr = Z s f ( ε + r ) | r α − ( r + ε ) α | dr = Z ε f ( ε + r ) | r α − ( r + ε ) α | dr + Z sε f ( ε + r ) | r α − ( r + ε ) α | dr =: A + A . We have, again from the univariate f ’s increasingness, and the limit lim r → f ( r ) = 0, A ≤ f (2 ε ) Z ε | r α − ( r + ε ) α | dr = cst · f (2 ε ) ε α +1 = o (cid:16) ε /m (cid:17) . For the other part of A , we need to use f ’s concavity at the point 2 ε in the interval [0 , ε + r ] (since ε + r > ε in this case), which implies f ( ε + r ) < f (2 ε ) ( ε + r ) / (2 ε ). Also using the mean-valuetheorem for the difference of negative cubes, we get A ≤ cst · ε Z sε f ( ε + r ) r α − dr ≤ cst · εf (2 ε ) Z sε ( ε + r ) r α − dr ≤ cst · εf (2 ε ) Z sε r α − = cst · ε α +1 f (2 ε ) = o (cid:16) ε / (cid:17) . This finishes the proof of Condition (20).
Step 2. Covariance calculation . We first calculate the second mixed derivative ∂ ˜ δ /∂s∂t , where ˜ δ is the canonical metric of Z , because we must show | µ | ( OD ) ≤ ε α , which is condition (21), and µ ( dsdt ) = ds dt ∂ ˜ δ /∂s∂t . We have, for 0 ≤ s ≤ t − ε ,˜ δ ( s, t ) = Z s ( g ( t, s − r ) − g ( s, s − r )) dr + Z ts g ( t, r ) dr =: A + B. We calculate ∂ A∂s∂t ( t, s ) = 2 ∂g∂t ( t,
0) ( g ( t, − g ( s, Z s ∂g∂t ( t, s − r ) (cid:18) ∂g∂s ( t, s − r ) − ∂g∂t ( s, s − r ) − ∂g∂s ( s, s − r ) (cid:19) + Z s g ( t, s − r ) − g ( s, s − r )) ∂ g∂s∂t ( t, s − r ) dr. = A + A + A , and ∂ B∂s∂t ( t, s ) = − g ( t, s ) ∂g∂t ( t, s ) . Next, we immediately get, for the portion of | µ | ( OD ) corresponding to B , using the hypothesesof our proposition, Z Tε dt Z t − ε ds (cid:12)(cid:12)(cid:12)(cid:12) ∂ B∂s∂t ( t, s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c Z Tε dt Z t − ε dsf ( | t − s | ) | t − s | α | t − s | α − ≤ c k f k ∞ Z Tε dt ε α = cst · ε α , A , using ourhypotheses, we have Z Tε dt Z t − ε ds | A | ≤ Z Tε dt Z t − ε ds t α (cid:12)(cid:12)(cid:12)(cid:12) ∂g∂t ( ξ t,s , (cid:12)(cid:12)(cid:12)(cid:12) | t − s | where ξ t,s is in the interval ( s, t ). Our hypothesis thus implies (cid:12)(cid:12)(cid:12) ∂g∂t ( ξ t,s , (cid:12)(cid:12)(cid:12) ≤ s α , and hence Z Tε dt Z t − ε ds | A | ≤ T Z Tε dt Z t − ε ds s α − t α − = 2 T α − Z Tε dt t α − ( t − ε ) α ≤ α − T α . This is much smaller than the right-hand side ε α of Condition (21), since 2 α = 1 /m − <
0. Theterms A and A are treated similarly, thanks to our hypotheses. Step 3: proving Condition (34).
In fact, we modify the proof of Theorem 12, in particular Steps 3and 4, so that we only need to prove Z Tt =2 ε dt Z t − εs =0 ds Z Tu =0 (cid:12)(cid:12)(cid:12) ∆ ˜ G t ( u ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∆ ˜ G s ( u ) (cid:12)(cid:12)(cid:12) du ≤ cε α = cε /m +1 , (37)instead of Condition (34). Indeed, for instance in Step 3, this new condition yields a final contri-bution of order δ m − ( ε ) ε − ε /m +1 . With the assumption on δ that we have, δ ( ε ) = o (cid:0) ε / (2 m ) (cid:1) ,and hence the final contribution is of order o (cid:0) ε (2 m − / (2 m ) − /m (cid:1) = o (1). This proves that theconclusion of Theorem 12 holds if we assume (37) instead of Condition (34).We now prove (37). We can write Z Tt =2 ε dt Z t − εs =0 ds Z Tu =0 (cid:12)(cid:12)(cid:12) ∆ ˜ G t ( u ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∆ ˜ G s ( u ) (cid:12)(cid:12)(cid:12) du = Z Tt =2 ε dt Z t − εs =0 ds Z s | g ( t + ε, u ) − g ( t, u ) | | g ( s + ε, u ) − g ( s, u ) | du + Z Tt =2 ε dt Z t − εs =0 ds Z s + εs | g ( t + ε, u ) − g ( t, u ) | | g ( s + ε, u ) | du =: A + B. For A , we use the hypotheses of this proposition: for the last factor in A , we exploit the factthat g is decreasing in t while f is increasing in t ; for the other factor in A , use the bound on ∂g/∂t ;thus we have A ≤ Z Tt =2 ε dt Z t − εs =0 ε | t − s | α − ds Z s f ( s + ε, u ) (( s − u ) α − ( s + ε − u ) α ) du. We separate the integral in u into two pieces, for u ∈ [0 , s − ε ] and u ∈ [ s − ε, s ]. For the firstintegral in u , since f is bounded, we have Z s − ε f ( s + ε, u ) (( s − u ) α − ( s + ε − u ) α ) du ≤ k f k ∞ ε Z s − ε ( s − u ) α − du ≤ k f k ∞ c α ε α . For the second integral in u , we use the fact that s − u + ε > ε and s − u < ε implies s − u + ε > s − u ), so that the negative part of the integral can be ignored, and thus Z ss − ε f ( s + ε, u ) (( s − u ) α − ( s + ε − u ) α ) du ≤ k f k ∞ Z ss − ε ( s − u ) α du = k f k ∞ c α ε α , u . Thus A ≤ cst · ε α Z Tt =2 ε dt Z t − εs =0 | t − s | α − ds ≤ cst · ε α Z Tt =2 ε dt ε α ≤ cst · ε α = cst · ε /m +1 , which is the conclusion we needed at least for A .Lastly, we estimate B . We use the fact that f is bounded, and thus | g ( s + ε, u ) | ≤ k f k ∞ | s + ε − u | α ,as well as the estimate on the derivative of g as we did in the calculation of A , yielding B ≤ k f k ∞ ε Z Tt =2 ε dt Z t − εs =0 ds | t − s − ε | α − Z s + εs | s + ε − u | α du = cst · ε α +2 Z Tt =2 ε dt Z t − εs =0 ds | t − s − ε | α − ≤ | α | cst · ε α +2 Z Tt =2 ε dt Z t − εs =0 ds | t − s | α − ≤ cst · ε α +2 = cst · ε /m +1 . This is the conclusion we needed for B, which finishes the proof of the proposition.The above proposition covers a wide variety of martingale-based models, which can be quitefar from Gaussian models in the sense that they can have only a few moments. We describe oneeasily constructed class. Assume that M is a martingale such that E h | d [ M ] /dt | m i is boundedabove by a constant c m uniformly in t ≤ T . This uniform boundedness assumption impliesthat we can take Γ ≡ c in Condition (A). In particular, ˜ G can be chosen to be proportional to G . Let G ( t, s ) = G RLfBm ( t, s ) := s ≤ t | t − s | / (2 m ) − / α for some α >
0; in other words, G is the Brownian representation kernel of the Riemann-Liouville fractional Brownian motion withparameter H = 1 / (2 m ) − α > / (2 m ). It is immediate to check that the assumptions of Proposition13 are satisfied for this class of martingale-based models, which implies that the corresponding X defined by (7) have zero m th variation.More generally, assume that G is bounded above by a multiple of G RLfBm , and assume thetwo partial derivatives of G , and the mixed second order derivative of G , are bounded by thecorresponding (multiples of) derivatives of G RLfBm ; one can check that the standard fBm’s kernelis in this class, and that the martingale-based models of this class also satisfy the assumptions ofProposition 13, resulting again zero m th variations for the corresponding X defined in (7). For thesake of conciseness, we will omit the details, which are tedious and straightforward.The most quantitatively significant condition in Theorem 12, that the univariate function δ ( ε )corresponding to ˜ G be equal to o (cid:0) ε / (2 m ) (cid:1) , can be interpreted as a regularity condition. In theGaussian case, it means that there is a function f ( ε ) = o (cid:16) ε / (2 m ) log / (1 /ε ) (cid:17) such that f isan almost-sure uniform modulus of continuity for X . In non-Gaussian cases, similar interpre-tations can be given for the regularity of X itself, provided enough moments of X exist. If X has fractional exponential moments, in the sense that for some constants c > , < β ≤ E h exp (cid:16) c | X ( t ) − X ( s ) | β (cid:17)i is finite for all s, t , then the function f above will also serve as analmost-sure uniform modulus of continuity for X , provided the logarithmic correction term in f israised to the power 1 /β rather than 1 /
2. Details of how this can be established are in the non-Gaussian regularity theory in [30]. If X has standard moments of all orders, then one can replace f ( ε ) by ε / (2 m ) − α for any α >
0. This is easily achieved using Kolmogorov’s continuity criterion. If X only has finitely many moments, Kolmogorov’s continuity criterion can only guarantee that one34ay take α greater than some α >
0. We do not delve into the details of these regularity issuesin the non-Gaussian martingale case.
In this section, we investigate the possibility of defining the so-called symmetric stochastic integraland its associated Itˆo formula for processes which are not fractional Brownian motion; fBm wastreated in [14]. We concentrate on Gaussian processes under hypotheses similar to those used inSection 3.3 (Theorem 8).The basic strategy is to use the results of [14]. Let X be a stochastic process on [0 , g on R , the limitlim ε → ε Z du ( X u + ε − X u ) m g (cid:18) X u + ε + X u (cid:19) = 0 (38)holds in probability, for both m = 3 and m = 5, then for every t ∈ [0 ,
1] and every f ∈ C ( R ), the symmetric (“generalized Stratonovich”) stochastic integral Z t f ′ ( X u ) d ◦ X u =: lim ε → ε Z t du ( X u + ε − X u ) 12 (cid:0) f ′ ( X u + ε ) + f ′ ( X u ) (cid:1) (39)exists and we have the Itˆo formula f ( X t ) = f ( X ) + Z t f ′ ( X u ) d ◦ X u . (40)Our goal is thus to prove (38) for a wide class of Gaussian processes X , which will in turn implythe existence of (39) and the Itˆo formula (40).If X has homogeneous increments in the sense of Section 3.2, meaning that E h ( X s − X t ) i = δ ( t − s ) for some univariate canonical metric function δ , then by using g ≡ and our Theorem 6,we see that for (38) to hold, we must have δ ( r ) = o (cid:0) r / (cid:1) . If one wishes to treat non-homogeneouscases, we notice that (38) for g ≡ δ ( r ) = o (cid:0) r / (cid:1) .But we will also need some non-degeneracy conditions in order to apply the quartic linear regressionmethod of [14]. These are Conditions (i) and (ii) in the next Theorem. Condition (iii) therein isessentially a consequence of the condition that δ be increasing and concave. These conditions areall further discussed after the statement of the next theorem and its corollary. Theorem 14
Let m ≥ be an odd integer. Let X be a Gaussian process on [0 , satisfying thehypotheses of Theorem 8. This means in particular that we denote as usual its canonical metricby δ ( s, t ) , and that there exists a univariate increasing and concave function δ such that δ ( r ) = o (cid:0) r / (2 m ) (cid:1) and δ ( s, t ) ≤ δ ( | t − s | ) . Assume that for u < v , the functions u V ar [ X u ] =: Q u , v δ ( u, v ) , and u
7→ − δ ( u, v ) are increasing and concave. Assume there exist positive constants a > , b < / , c > / , and c ′ > such that for all ε < u < v ≤ , (i) cδ ( u ) ≤ Q u , (ii) c ′ δ ( u ) δ ( v − u ) ≤ Q u Q v − Q ( u, v ) , iii) δ ( au ) − δ ( u )( a − u < b δ ( u ) u . (41) Then for every bounded measurable function g on R , lim ε → ε E "(cid:18)Z du ( X u + ε − X u ) m g (cid:18) X u + ε + X u (cid:19)(cid:19) = 0 . When we apply this theorem to the case m = 3, the assumption depending on m , namely δ ( r ) = o (cid:0) r / (2 m ) (cid:1) is satisfied a fortiori for m = 5 as well, which means that under the assumption δ ( r ) = o (cid:0) r / (cid:1) , the theorem’s conclusion holds for m = 3 and m = 5. Therefore, as mentioned inthe strategy above, we immediately get the following. Corollary 15
Assume the hypotheses of Theorem 14 with m = 3 . We have existence of thesymmetric integral in (39), and its Itˆo formula (40), for every f ∈ C ( R ) and t ∈ [0 , . Before proceeding to the proof of this theorem, we discuss its hypotheses. We refer to thedescription at the end of Section 3.3 for examples satisfying the hypotheses of Theorem 8; theseexamples also satisfy the monotonicity and convexity conditions in the above theorem.Condition (i) is a type of coercivity assumption on the non-degeneracy of X ’s variances incomparison to its increments’ variances. The hypotheses of Theorem 8 imply that Q u ≤ δ ( u ), andCondition (i) simply adds that these two quantities should be commensurate, with a lower boundthat it not too small. The ”Volterra convolution”-type class of processes (26) given at the end ofSection 3.3, which includes the Riemann-Liouville fBm’s, satisfies Condition (i) with c = 1 /
2. Inthe homogeneous case, (i) is trivially satisfied since Q u ≡ δ ( u ).Condition (ii) is also a type of coercivity condition. It too is satisfied in the homogeneouscase. We prove this claim, since it is not immediately obvious. In the homogeneous case, since δ ( u, v ) = δ ( v − u ) = Q v − u , we calculate Q u Q v − Q ( u, v ) = Q u Q v − − ( Q u + Q v − Q v − u ) and after rearranging some terms we obtain Q u Q v − Q ( u, v ) = 2 − Q v − u ( Q u + Q v ) − − ( Q v − Q u ) − − Q v − u . We note first that by the concavity of Q , we have Q v − Q u < Q v − u , and consequently, ( Q v − Q u ) ≤ ( Q v − Q u ) Q v − u ≤ Q v Q v − u . This implies Q u Q v − Q ( u, v ) ≥ − Q v − u Q u + 4 − (cid:0) Q v − u Q v − Q v − u (cid:1) . Now by monotonicity of Q , we can write Q v − u Q v ≥ Q v − u . This, together with Condition (i), yieldCondition (ii) since we now have Q u Q v − Q ( u, v ) ≥ − Q v − u Q u ≥ − c δ ( v − u ) δ ( u ) . Lastly, Condition (iii) represents a strengthened concavity condition on the univariate function δ . Indeed, the left-hand side in (41) is the slope of the secant of the graph of δ between the points u and au , while the right-hand side is b times the slope of the secant from 0 to u . If b were allowed36o be 1, (iii) would simply be a consequence of convexity. Here taking b ≤ / δ ; the fact that condition (iii) requires slightly more, namely b strictlyless than 1 /
2, allows us to work similarly to the scale δ ( r ) = r H with H < /
2, as opposed tosimply asking H ≤ /
2. Since the point of the Theorem is to allow continuity moduli which arearbitrarily close to r / , Condition (iii) is hardly a restriction. Proof of Theorem 14.
Step 0: setup.
The expectation to be evaluated is written, as usual, as a double integral over( u, v ) ∈ [0 , . For ε > D ε = (cid:8) ( u, v ) ∈ [0 , : ε − ρ ≤ u ≤ v − ε − ρ < v ≤ (cid:9) where ρ ∈ (0 ,
1) is fixed. Using the boundedness of g and Cauchy-Schwarz’s inequality, thanks tothe hypothesis δ ( r ) = o (cid:0) r / (2 m ) (cid:1) , the term corresponding to the diagonal part (integral over D cε )can be treated identically to what was done in [14] in dealing with their term J ′ ( ε ) following thestatement of their Lemma 5.1, by choosing ρ small enough. It is thus sufficient to prove that J ( ε ) := 1 ε E (cid:20)Z Z D ε dudv ( X u + ε − X u ) m ( X v + ε − X v ) m g (cid:18) X u + ε + X u (cid:19) g (cid:18) X v + ε + X v (cid:19)(cid:21) tends to 0 as ε tends to 0. We now use the same method and notation as in Step 3 of the proofof Theorem 4.1 in [14]. In order to avoid repeating arguments from that proof, we only state andprove the new lemmas which are required. Step 1: translating Lemma 5.3 from [14].
Using the fact that E (cid:2) Z ℓ (cid:3) ≤ E (cid:2) G ℓ (cid:3) ≤ δ ( ε ), this lemmatranslates as: Lemma 16
Let k ≥ be an integer. Then Z Z D ε E h | Γ ℓ | k i dudv ≤ cst · εδ k ( ε ) . This step and the next 4 steps are devoted to the
Proof of lemma 16.
We only need to showthat for all i, j ∈ { , } , Z Z D ε | r ij | k dudv ≤ cst · εδ k ( ε ) . (42)Recall the function K defined in [14] K ( u, v ) := E [( X u + ε + X u ) ( X v + ε + X v )]= Q ( u + ε, v + ε ) + Q ( u, v + ε ) + Q ( u + ε, v ) + Q ( u, v ) . This is not to be confused with the usage of the letter K in previous sections, to which there willbe made no reference in this proof; the same remark hold for the notation ∆ borrowed again from[14], and used below.To follow the proof in [14], we need to prove the following items for some constants c and c :1. c δ ( u ) ≤ K ( u, u ) ≤ c δ ( u ) ; 37. K ( u, v ) ≤ c δ ( u ) δ ( v ) ;3. ∆ ( u, v ) := K ( u, u ) K ( v, v ) − K ( u, v ) ≥ c δ ( u ) δ ( v − u ) . By the Theorem’s upper bound assumption on the bivariate δ (borrowed from Theorem 8),its assumptions on the monotonicity of Q and the univariate δ , and finally using the coercivityassumption (i), we have K ( u, u ) = Q u + Q u + ε + 2 Q ( u, u + ε ) = 2 ( Q u + Q u + ε ) − δ ( u, u + ε ) ≥ Q u + Q u + ε ) − δ ( ε ) ≥ Q u − δ ( ε ) ≥ (cid:0) − c − (cid:1) Q u . This proves the lower bound in Item 1 above. The upper bound in Item 1 is a special case ofItem 2, which we now prove. Again, the assumption borrowed from Theorem 8, which says that δ ( s, t ) ≤ δ ( | t − s | ), now implies, for s = 0, that δ (0 , u ) = Q u ≤ δ ( u ) . (43)We write, via Cauchy-Schwarz’s inequality and the fact that δ is increasing, and thanks to (43), K ( u, v ) ≤ δ ( u + ε ) δ ( v + ε ) . However, since δ is concave with δ (0) = 0, we have δ (2 u ) / u ≤ δ ( u ) /u . Also, since we are inthe set D ε , u + ε ≤ u and v + ε ≤ v . Hence K ( u, v ) ≤ δ (2 u ) δ (2 v ) ≤ δ ( u ) δ ( v ) , which is Item 2.We now verify Item 3 for all u, v ∈ D ε , assuming in addition that v is not too small, specifically v > ε ρ/ . One can estimate the integral in Lemma 16 restricted to those values where v ≤ ε ρ/ using coarser tools than we use below; we omit the corresponding calculations. From the definitionof K above, using the fact that, by our concavity assumptions, Q is, in both variables, a sum ofLipschitz functions, we have, for small ε , K ( u, v ) = 4 Q ( u, v ) + O ( ε ) . Therefore, ∆ = 16 (cid:0) Q u Q v − Q ( u, v ) (cid:1) + O ( ε ) . Assumption (ii) in the Theorem now implies∆ ≥ c ′ δ ( u ) δ ( v − u ) + O ( ε ) . The concavity of Q and Assumption (i) imply δ ( r ) ≥ Q r ≥ cst · r . Moreover, because of therestriction on v , either v − u > cst · ε ρ/ or u > cst · ε ρ/ . Therefore δ ( u ) δ ( v − u ) ≥ cst · ε − ρ ε ρ/ ≫ ε . Therefore, for ε small enough, ∆ ≥ c ′ δ ( u ) δ ( v − u ), proving Item 3.38t will now be necessary to reestimate the components of the matrix Λ where we recallΛ [11] := E [( X u + ε + X u ) ( X u + ε − X u )] , Λ [12] := E [( X v + ε + X v ) ( X u + ε − X u )] , Λ [21] := E [( X u + ε + X u ) ( X v + ε − X v )] , Λ [22] := E [( X v + ε + X v ) ( X v + ε − X v )] . Step 2: the term r . We have by the lower bound of item 1 above on K ( u, u ), | r | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p K ( u, u ) Λ [11] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ cstδ ( u ) | Λ [11] | . To bound | Λ [11] | above, we write | Λ [11] | = | E [( X u + ε + X u ) ( X u + ε − X u )] | = Q u + ε − Q u ≤ εQ ( u ) /u ≤ εδ ( u ) /u where we used the facts that Q u is increasing and concave, and that Q u ≤ δ ( u ). Thus we have | r | ≤ ε cst δ ( u ) u . The result (42) for i = j = 1 now follows by the next lemma. Lemma 17
For every k ≥ , there exists c k > such that for every ε ∈ (0 , , Z ε (cid:12)(cid:12)(cid:12)(cid:12) δ ( u ) u (cid:12)(cid:12)(cid:12)(cid:12) k du ≤ c k ε (cid:12)(cid:12)(cid:12)(cid:12) δ ( ε ) ε (cid:12)(cid:12)(cid:12)(cid:12) k . Proof of lemma 17.
Our hypothesis (iii) can be rewritten as δ ( au ) au < (cid:18) a − ba (cid:19) δ ( u ) u =: K a,b δ ( u ) u . The concavity of δ also implies that δ ( u ) /u is increasing. Thus we can write Z ε (cid:12)(cid:12)(cid:12)(cid:12) δ ( u ) u (cid:12)(cid:12)(cid:12)(cid:12) k du ≤ ∞ X j =0 Z εa j +1 εa j (cid:12)(cid:12)(cid:12)(cid:12) δ ( u ) u (cid:12)(cid:12)(cid:12)(cid:12) k du ≤ ∞ X j =0 (cid:0) εa j +1 − εa j (cid:1) | K a,b | jk (cid:12)(cid:12)(cid:12)(cid:12) δ ( ε ) ε (cid:12)(cid:12)(cid:12)(cid:12) k = ε ( a − (cid:12)(cid:12)(cid:12)(cid:12) δ ( ε ) ε (cid:12)(cid:12)(cid:12)(cid:12) k ∞ X j =0 (cid:16) | K a,b | k a (cid:17) j . f ( a ) := | K a,b | k a < a >
1. We have f (1) = 0 and f ′ (1) = k (1 − b ) −
1. This last quantity is strictly positive for all k ≥ b < /
2. This finishes the proof of the lemma 17. (cid:3)
Step 3: the term r . We have r = Λ [11] − K ( u, v ) p K ( u, u ) ∆ ( u, v ) + Λ [12] p K ( u, u ) p ∆ ( u, v ) . We saw in the previous step that | Λ [11] | = | Q u + ε − Q u | ≤ cst · εδ ( u ) /u . For Λ [12], using thehypotheses on our increasing and concave functions, we calculate | Λ [12] | = (cid:12)(cid:12) Q u + ε − Q u ) + δ ( u + ε, v + ε ) − δ ( u, v + ε ) + δ ( u + ε, v ) − δ ( u, v ) (cid:12)(cid:12) ≤ | Λ [11] | + εδ ( u + ε, v + ε ) / ( v − u ) + εδ ( u + ε, v ) / ( v − u − ε ) ≤ | Λ [11] | + εδ ( v − u ) / ( v − u ) + εδ ( v − u − ε ) / ( v − u − ε ) ≤ cst · εδ ( u ) /u + 2 εδ ( v − u − ε ) / ( v − u − ε ) . (44)The presence of the term − ε in the last expression above is slightly aggravating, and one wouldlike to dispose of it. However, since ( u, v ) ∈ D ε , we have v − u > ε ρ for some ρ ∈ (0 , v − u − ε > ε ρ − ε > ε ρ/ for ε small enough. Hence by using ρ/ ρ in the definition of D ε in the current calculation, we can ignore the term − ε in the last displayed line above. Togetherwith items 1, 2, and 3 above which enable us to control the terms K and ∆ in r , we now have | r | ≤ cst · ε δ ( u ) u (cid:18) δ ( u ) δ ( v ) δ ( u ) δ ( u ) δ ( v − u ) + δ ( u ) δ ( u ) δ ( v − u ) (cid:19) + cst · ε δ ( v − u ) v − u δ ( u ) δ ( u ) δ ( v − u )= cst · ε (cid:18) δ ( u ) δ ( v ) uδ ( v − u ) + δ ( u ) uδ ( v − u ) + δ ( v − u ) v − u (cid:19) . We may thus write
Z Z D ε | r | k dudv ≤ cst · ε k Z Z D ε (cid:12)(cid:12)(cid:12)(cid:12) δ ( u ) δ ( v ) uδ ( v − u ) (cid:12)(cid:12)(cid:12)(cid:12) k + (cid:12)(cid:12)(cid:12)(cid:12) δ ( u ) uδ ( v − u ) (cid:12)(cid:12)(cid:12)(cid:12) k + (cid:12)(cid:12)(cid:12)(cid:12) δ ( v − u ) v − u (cid:12)(cid:12)(cid:12)(cid:12) k ! dudv. The last term RR D ε (cid:12)(cid:12)(cid:12) δ ( v − u ) v − u (cid:12)(cid:12)(cid:12) k dudv is identical, after a trivial change of variables, to the one dealtwith in Step 2. Since δ is increasing, second the term RR D ε (cid:12)(cid:12)(cid:12) δ ( u ) uδ ( v − u ) (cid:12)(cid:12)(cid:12) k dudv is smaller than the firstterm RR D ε (cid:12)(cid:12)(cid:12) δ ( u ) δ ( v ) uδ ( v − u ) (cid:12)(cid:12)(cid:12) k dudv . Thus we only need to deal with that first term; it is more delicate thanwhat we estimated in Step 2.We separate the integral over u at the intermediate point v/
2. When u ∈ [ v/ , v − ε ], we usethe estimate δ ( u ) u ≤ δ ( v/ v/ ≤ δ ( v ) v .
40n the other hand when u ∈ [ ε, v/
2] we simply bound 1 /δ ( v − u ) by 1 /δ ( v/ Z Z D ε (cid:12)(cid:12)(cid:12)(cid:12) δ ( u ) δ ( v ) uδ ( v − u ) (cid:12)(cid:12)(cid:12)(cid:12) k dudv = Z v =2 ε dv Z v/ u = ε (cid:12)(cid:12)(cid:12)(cid:12) δ ( u ) δ ( v ) uδ ( v − u ) (cid:12)(cid:12)(cid:12)(cid:12) k du + Z v = ε dv Z v − εu = v/ (cid:12)(cid:12)(cid:12)(cid:12) δ ( u ) δ ( v ) uδ ( v − u ) (cid:12)(cid:12)(cid:12)(cid:12) k du ≤ Z v =2 ε dv (cid:12)(cid:12)(cid:12)(cid:12) δ ( v ) δ ( v/ (cid:12)(cid:12)(cid:12)(cid:12) k Z v/ u = ε (cid:12)(cid:12)(cid:12)(cid:12) δ ( u ) u (cid:12)(cid:12)(cid:12)(cid:12) k du + 2 Z v = ε (cid:12)(cid:12)(cid:12)(cid:12) δ ( v ) v (cid:12)(cid:12)(cid:12)(cid:12) k dv Z v − εu = v/ (cid:12)(cid:12)(cid:12)(cid:12) δ ( v − u ) (cid:12)(cid:12)(cid:12)(cid:12) k du ≤ k Z u = ε (cid:12)(cid:12)(cid:12)(cid:12) δ ( u ) u (cid:12)(cid:12)(cid:12)(cid:12) k du + 2 1 δ k ( ε ) Z v = ε v k (cid:12)(cid:12)(cid:12)(cid:12) δ ( v ) v (cid:12)(cid:12)(cid:12)(cid:12) k dv ≤ cst · ε (cid:18) δ ( ε ) ε (cid:19) k ;here we used the concavity of δ to imply that δ ( v ) /δ ( v/ ≤
2, and to obtain the last line, weused Lemma 17 for the first term in the previous line, and we used the fact that δ is increasing andthat v ≤
1, together again with Lemma 17 for the second term in the previous line. This finishesthe proof of (42) for r . Step 4: the term r . We have r = Λ [21] 1 p K ( u, u )and similarly to the previous step, | Λ [21] | = | Q ( u + ε, v + ε ) − Q ( u + ε, v ) + Q ( u, v + ε ) − Q ( u, v ) | = (cid:12)(cid:12) Q v + ε − Q v ) + δ ( u + ε, v ) − δ ( u + ε, v + ε ) + δ ( u, v ) − δ ( u, v + ε ) (cid:12)(cid:12) ≤ | Λ [11] | + ε δ ( u + ε, v ) v − u − ε + ε δ ( u, v ) v − u ≤ cst · εδ ( u ) /u + 4 εδ ( v − u ) / ( v − u ) , which is the same expression as in (44). Hence with the lower bound of Item 1 on K ( u, u ) we have Z Z D ε | r | k dudv ≤ cst · ε k Z Z D ε (cid:12)(cid:12)(cid:12)(cid:12) δ ( u ) u (cid:12)(cid:12)(cid:12)(cid:12) k + (cid:12)(cid:12)(cid:12)(cid:12) δ ( v − u )( v − u ) δ ( u ) (cid:12)(cid:12)(cid:12)(cid:12) k ! dudv = cst · ε k Z Z D ε (cid:12)(cid:12)(cid:12)(cid:12) δ ( u ) u (cid:12)(cid:12)(cid:12)(cid:12) k + (cid:12)(cid:12)(cid:12)(cid:12) δ ( u ) uδ ( v − u ) (cid:12)(cid:12)(cid:12)(cid:12) k ! dudv. This is bounded above by the expression obtained as an upper bound in Step 3 for RR D ε | r | k dudv ,which finishes the proof of (42) for r . Step 5: the term r . Here we have r = Λ [21] − K ( u, v ) p K ( u, u ) ∆ ( u, v ) + Λ [22] p K ( u, u ) p ∆ ( u, v ) .
41e have already seen in the previous step that | Λ [21] | ≤ cst · ε (cid:18) δ ( u ) u + δ ( v − u ) v − u (cid:19) . Moreover, we have, as in Step 2, | Λ [22] | = | Q v + ε − Q v | ≤ cst · ε δ ( v ) v . Thus using the bounds in items 1, 2, and 3, | r | ≤ cst · ε (cid:20)(cid:18) δ ( u ) u + δ ( v − u ) v − u (cid:19) δ ( u ) δ ( v ) δ ( u ) δ ( v − u ) + δ ( v ) v δ ( u ) δ ( u ) δ ( v − u ) (cid:21) = cst · ε (cid:20) δ ( u ) δ ( v ) uδ ( v − u ) + δ ( v ) δ ( v − u ) δ ( u ) ( v − u ) + δ ( v ) vδ ( v − u ) (cid:21) . Of the last three terms, the first term was already treated in Step 3, the second is, up to a changeof variable, identical to the first, and the third is smaller than δ ( u ) uδ ( v − u ) which was also treated inStep 3. Thus (42) is proved for r , which finishes the entire proof of Lemma 16. (cid:3) Step 6: translating Lemma 5.4 from [14].
We will prove the following result
Lemma 18
For all j ∈ { , , · · · , ( m − / } , Z Z D ε | E [ Z Z ] | m − j dudv ≤ cst · εδ m − j ) ( ε ) . Proof of Lemma 18.
As in [14], we have | E [ Z Z ] | m − j ≤ cst · | E [ G G ] | m − j + cst · | E [Γ Γ ] | m − j . The required estimate for the term corresponding to | E [Γ Γ ] | m − j follows by Cauchy-Schwarz’s in-equality and Lemma 16. For the term corresponding to | E [ G G ] | m − j , we recognize that E [ G G ]is the negative planar increment Θ ε ( u, v ) defined in (12). Thus the corresponding term was alreadyconsidered in the proof of Theorem (8). More specifically, up to the factor ε δ − j ( ε ), we now haveto estimated the same integral as in Step 2 of that theorem’s proof: see expression (24) for theterm we called J j,OD . This means that Z Z D ε | E [ G G ] | m − j dudv ≤ ε δ j ( ε ) J j,OD ≤ ε | µ | ( OD ) δ m − j − ( ε ) . Our hypotheses borrowed from Theorem (8) that | µ | ( OD ) ≤ cst · ε /m − and that δ ( ε ) = o (cid:0) r / (2 m ) (cid:1) now imply that the above is ≪ εδ m − j ) ( ε ), concluding the lemma’s proof. (cid:3) Step 7. Conclusion . The remainder of the proof of the theorem is to check that Lemmas 16 and 18do imply the claim of the theorem; this is done exactly as in Steps 3 and 4 of the proof of Theorem4.1 in [14]. Since such a task is only bookkeeping, we omit it, concluding the proof of Theorem 14. (cid:4)
Acknowledgements
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