A new inequality for the Riemann-Stieltjes integrals driven by irregular signals in Banach spaces

We prove an inequality of the Loéve-Young type for the Riemann-Stieltjes integrals driven by irregular signals attaining their values in Banach spaces, and, as a result, we derive a new theorem on the existence of the Riemann-Stieltjes integrals driven by such signals. Also, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p\ge1$\end{document}p≥1, we introduce the space of regulated signals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f:[a,b]\rightarrow W$\end{document}f:[a,b]→W (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a< b$\end{document}a<b are real numbers, and W is a Banach space) that may be uniformly approximated with accuracy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\delta>0$\end{document}δ>0 by signals whose total variation is of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\delta^{1-p}$\end{document}δ1−p as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\delta\rightarrow0+$\end{document}δ→0+ and prove that they satisfy the assumptions of the theorem. Finally, we derive more exact, rate-independent characterisations of the irregularity of the integrals driven by such signals.


Introduction
The first aim of this paper is a generalisation of the results of [1] and [2] to the functions attaining their values not only in R but in more general spaces. Next, to obtain more precise results, for any p ≥ 1, we introduce the space U p ([a, b], W ) of regulated functions/signals f : [a, b] → W (a < b are real numbers, and W is a Banach space) that may be uniformly approximated with accuracy δ > 0 by functions whose total variation is of order δ 1-p as δ → 0+. This way we will obtain a result about the existence of the limit of Riemann-Stieltjes sums, which we will denote by b a f dg, for functions from U p ([a, b]) and U q ([a, b]) whenever p, q > 1, p -1 + q -1 > 1. Results of this type were earlier obtained by Young [3,4] and D'yačkov [5] (for very detailed account, see [6,Chapter 3]), but they were expressed in terms of p-or (more general) φ-variations. The integral obtained as the limit of Riemann-Stieltjes sums in this case is often called the Young integral. It is in place to mention that nowadays there exists fairly rich literature on the integration with respect to irregular integrators where the object defined as an integral is not necessarily obtained as the limit of Riemann-Stieltjes sums. In [7] an interesting approach based on fractional calculus was developed. In the modern theory of rough paths developed by Terry Lyons, the existence of integrals of the form values of the iterated integrals t 1 <s<t 2 dx i s , t 1 <s<t<t 2 dx i s dx j t , t 1 <s<t<u<t 2 dx i s dx j t dx k u , etc.; see [8,9]. Both approaches mentioned provide tools to deal also with the case where the integrand and integrator reveal the same irregularity as the standard Brownian motion or more general semimartingales (although the Young integral fails to exist in such case, and it was, historically, one of the main reasons for the development of the stochastic integral). To deal with the integrals driven by a fractional Brownian motion, which fails to be a semimartingale except the special case when its Hurst parameter is 1/2, one uses various integrals: Young's integral, fractional integral or the Skorohod integral; see [10,11].
To obtain the convergence of Riemann-Stieltjes sums, we will use a partial solution of a variational problem similar to that considered in [1]. In [1] the following problem was considered: given real a < b, c > 0, a regulated function/signal f : and the following bounds hold: where [1,Thm. 4 and Rem. 15]). Moreover, we have Unfortunately, this result is no more valid for functions attaining their values in more general metric spaces.

Remark
Remark 1 answers (negatively) the question posed few years ago by Krzysztof Oleszkiewicz if the truncated variation is the greatest lower bound for the total variation of functions from B(f , c/2) attaining values in R d , d = 2, 3, . . . , or in other spaces than R. Fortunately, it is possible to state an easy estimate of the left side of (2) in terms of the truncated variation of f , for f attaining values in any metric space (to define the total variation and the truncated variation of f attaining its values in the metric space (E, d), we just replace One application of Theorem 1 will be a generalisation of the results of [2] on the existence of the Riemann-Stieltjes integral. We will consider the case where the integrand and integrator attain their values in Banach spaces. The restriction to the Banach spaces stems from the fact that the method of our proof requires multiple application of summation by parts and proceeding to the limit of a Cauchy sequence, which may be done in a straightforward way in any Banach space. This way we will obtain a general theorem on the existence of the Riemann-Stieltjes integral along a path in some Banach space (E, · E ) (with the integrand being a path in the space L(E, V ) of continuous linear mappings F : E → V , where V is another Banach space) and an improved version of the Loéve-Young inequality for integrals driven by irregular paths in this space.
The famous Loéve-Young inequality may be stated as follows. If f : [a, b] → L(E, V ) and g : [a, b] → E are two regulated functions with no common points of discontinuity and f and g have finite p-and q-variations, respectively, where p > 1, q > 1 and p -1 + q -1 > 1, then the Riemann-Stieltjes integral b a f dg exists, and we have the following estimate: Here and denote the p-and q-variations of f and g, respectively (sometimes called the strong variations). The original Loéve-Young estimate with the constantC p,q = 1 + ζ (1/p + 1/q), where ζ is the famous Riemann zeta function, was formulated for real functions in [3]. The counterpart of this inequality for more general, Banach space-valued functions, with the con-stantC p,q = 4 1/p+1/q ζ (1/p + 1/q), is formulated in the proof of [12,Theorem 1.16]. Our improved version of (3) is the following: where f osc,[a,b] := sup a≤s<t≤b f (s)f (t) L(E,V ) , and C p,q is a universal constant depending on p and q only. Notice that always Let us comment shortly on the proofs of (3) and related results that have appeared so far. Young's original proof of (3) utlilised elementary but clever induction argument for finite sequences. Since then, there appeared several generalizations of (3), for example, based on control functions; see [8,Section 3.3] or [12,Section 1.3]. Another proof based on control functions but with different constant may be found in [13,Chapter 6]. An abstract version of the inequality proven in [13], called the sewing lemma, was proven in [14]. All these approaches give an inequality where only p-and q-variation (or Hölder) norms appear, whereas our approach gives an estimate where the p-variation norm of f , that is, [a,b] , which may be many times smaller than this norm. We formulate an even more improved version of the Loéve-Young inequality in Remark 2. The inequality formulated in Remark 2, together with relation (18), yields (4), and this, together with (5), yields (3), but reasoning in opposite direction (derivation of (4) or inequality stated in Remark 2 from (3)) seems to be not possible.
These results may be applied, for example, when f and g are trajectories of α-stable processes X 1 , X 2 with α ∈ (1, 2). However, since the obtained results are formulated in terms of rate-independent functionals, like the truncated variation or p-variation, they remain valid when f (t) = F(X 1 (A(t))) and g(t) = G(X 2 (B(t))) (with the technical assumption that the jumps of f and g do not occur at the same time), where A, B : [0, +∞) → [0, +∞) are piecewise monotonic, possibly random, changes of time (i.e. there exist 0 = T 0 < T 1 < · · · such that T n → +∞ almost surely as n → +∞, and A and B are monotonic on each interval (T i-1 , T i ), i = 1, 2, . . .), and F, G : R → R are locally Lipschitz.
As it was already mentioned, it appears that it is possible to derive weaker conditions under which the improved Loéve-Young inequality still holds, and we will prove that it still holds (and the Riemann-Stieltjes integral b a f dg exists) for functions f and g with no common points of discontinuity, satisfying respectively. Moreover, in such a case the indefinite integral I(t) := t a f dg reveals similar irregularity as the integrator g, namely, sup δ>0 δ q-1 TV δ (I, [a, b]) < ∞. We will also prove that, for any p ≥ 1, the class of functions f : is weaker than the finiteness of p-variation but stronger than the finiteness of q-variation for all q > p.
From early work of Lyons [15] it is well known that whenever f and g have finite p-and q-variations, respectively, p > 1, q > 1 and p -1 + q -1 > 1, then the indefinite integral I(·) has finite q-variation. However, it is also well known that a symmetric α-stable process X with α ∈ [1, 2] has finite p-variation for any p > α whereas its α-variation is infinite (on any proper compact subinterval of [0, +∞)); see, for example, [16,Thm. 4.1]. Thus, if, for example, f (t) = F(X 1 (A(t))) and g(t) = G(X 2 (B(t))) are like in the former paragraph, then we can say that I(·) has finite p-variation on any compact subinterval of [0, +∞) for any p > α but cannot say much more. From our results it will follow that I(·) ∈ U α ([0, t], R) for any t ≥ 0; moreover, we will get estimates for I stronger than those already known; see Theorem 3. Thus, the introduction of the new spaces and inequalities has at least two applications: (1) we identify the family of functions with finite p-variation as a proper subset of U p , that is, the family of functions that can be uniformly approximated with accuracy δ by simple functions whose total variation is of order δ 1-p as δ → 0+, (2) we get better estimates for the integrals driven by irregular paths, which may be used to strengthen some results on the existence of solutions of the differential equations driven by irregular paths; see [2, Section 3] and Section 4.3.
Let us comment on the organization of the paper. In the next section we prove very gen- where E is any metric space, in terms of the truncated variation of f . Next, in Section 3, we use the obtained estimates to prove a new theorem on the existence of the Riemann-Stieltjes integral driven by irregular paths in Banach spaces. In the proofs we closely follow [2]. In Section 4, we introduce the Banach spaces U p ([a, b], W ), p ≥ 1 (Section 4.1) and in Section 4.2 obtain more exact estimates of the rate-independent irregularity of functions from these spaces (in terms of φ-variation). In the last subsection we deal with the irregularity of the integrals driven by signals from the spaces U p ([a, b], W ), p ≥ 1.

Estimates for the variational problem
Let (E, d) be a metric space with metric d. For given reals a < b, we say that the function If E is complete, then a necessary and sufficient condition for f to be regulated is that it is a uniform limit of step functions (see [17,Thm. 7.6.1]). Let ) ≤ c/2. We will be interested in the following variational problem: find where To state our first main result, let us define the truncated variation of g with the truncation parameter c ≥ 0: Intuitively, the truncated variation with the truncation parameter c takes into account only those changes in the values of g whose distance is greater than c.
, f c (t)) ≤ c/2 and for any λ > 1, Thus the following estimates hold: In particular, taking λ = 2, we get the double-sided estimate The estimate from above follows from the following greedy algorithm. Let us consider the sequence of times defined in the following way: τ 0 = a and, for n = 1, 2, . . . , Note that, since f is regulated, lim n→+∞ τ n = +∞. (We apply the convention that inf ∅ = +∞.) Now we define a step function f c ∈ B(f , c/2) in the following way. For each n = 1, 2, . . .

Integration of irregular signals in Banach spaces
Directly from the definition it follows that the truncated variation is a superadditive functional of the interval, that is, for c ≥ 0 and d ∈ (a, b), Moreover, if (E, · E ) is a normed vector space (with norm · E ), then we also have the following easy estimate of the truncated variation of a function g : [a, b] → E perturbed by some other function h : [a, b] → E: which stems directly from the inequality Let now (E, · E ), (W , · W ) be Banach spaces, let (V , · V ) be another Banach space, and let (L(E, V ), · L(E,V ) ) be the space of continuous linear mappings F : E → V with the norm F L(E,V ) = sup e∈E: e E =1 F · e V . Throughout the rest of this paper, we will assume that f : [a, b] → W and g : [a, b] → E. We will often encounter the situation where W = L(E, V ).
Relations (11) and (12), together with Theorem 1, will allow us to establish a new result on the existence of the Riemann-Stieltjes integral with the integrand f : [a, b] → L(E, V ) and the integrator g. By saying that the Riemann-Stieltjes integral exists we will mean that for any sequence of partitions π n = {a = t n 0 < t n 1 < · · · < t n k(n) = b} such that mesh(π n ) := max i=1,2,...,k(n) (t n it n i-1 ) → 0 as n → +∞ and for any ξ n i ∈ [t n i-1 , t n i ], the Riemann-Stieltjes sums → E be two regulated functions that have no common points of discontinuity. Let η 0 ≥ η 1 ≥ · · · and θ 0 ≥ θ 1 ≥ · · · be two sequences of positive numbers such that η k ↓ 0, θ k ↓ 0 as k → +∞. Define η -1 := 1 2 sup a≤t≤b f (t)f (a) L(E,V ) and The proof of Theorem 2 is based on the following lemmas.
Now we proceed to the proof of Theorem 2.
Proof of Theorem 2 Again, the proof goes exactly along the same lines as the proof of [2, Thm. 1] with the obvious changes.

An improved version of the Loéve-Young inequality
Now we will obtain an improved version of the Loéve-Young inequality for integrals driven by irregular signals attaining their values in Banach spaces. Our main tool will be Theorem 2 and the following simple relation between the rate of growth of the truncated variation and finiteness of p-variation. If V p (f , [a, b]) < +∞ for some p ≥ 1, then for every δ > 0, This result follows immediately from the elementary estimate: for any x ≥ 0,  Proof By Theorem 2 it suffices to prove that, for some positive sequences η 0 ≥ η 1 ≥ · · · and θ 0 ≥ θ 1 ≥ · · · such that η k ↓ 0, θ k ↓ 0 as k → +∞, and η -1 = sup a≤t≤b f (t)f (a) L(E,V ) , we have The proof will follow from the proper choice of the sequences (η k ) and (θ k ). Choose and for k = 0, 1, . . . , define and θ k = γ · 3 -(α 2 /[(q-1)(p-1)]) k α/(q-1) .

U p ([a, b], W) as a Banach space
Let p ≥ 1, and let W be a Banach space. In this subsection, we will prove that the fam- , [a, b]) < +∞ is a Banach space and that the functional is a seminorm on this space (whereas the functional f TV,p,[a,b] = f (a) W + f p-TV,[a,b] is a norm). From (18) it follows that and ). It appears that this inclusion is strict. For example, if 0 ≤ a < b, then a real symmetric α-stable process X with α ∈ (1, 2] has finite p-variation for p > α, whereas (as it was already mentioned in the Introduction) its α-variation is a.s. infinite (on any proper compact subinterval of [0, +∞)). On the other hand, trajectories of X belong a.s. to U α ([0, t], R) for any t ≥ 0; see [18]. For another example, see [19,Thm. 17].
From the results of the next subsection it will also follow that but, again, this inclusion is strict.

Remark 2
For further justifcation of the importance of the spaces U p ([a, b], W ), p > 1, let us also notice that if W = L(E, V ), f belongs to U q ([a, b], W ), g belongs to U q ([a, b], E) for some q > 1 such that p -1 + q -1 > 1, and f and g have no common points of discontinuity, then the integral b a f dg still exists, and we have the estimate [a,b] with the same constant C p,q that appears in Corollary 1. This follows from the fact that in the proof of Corollary 1 we were using only estimate (18), which now may be replaced by the estimate for any δ > 0, stemming directly from the definition of the norm · p-TV, [a,b] .  ([a, b], W ) is simply the same as the space of functions with bounded total variation. Therefore, for the rest of the proof, we will assume that p > 1.

Proposition 1 For any p ≥ 1, the functional · p-TV,[a,b] is a seminorm, and the functional
The homogeneity of · p-TV, [a,b] and · TV,p,[a,b] follows easily from the fact that, for , which is a consequence of the equality To prove the triangle inequality, let us take f , h ∈ U p ([a, b]) and fix ε > 0. Let δ 0 > 0 and a ≤ t 0 < t 1 < · · · < t n ≤ b be such that where (·) + := max{·, 0}. By standard calculus, for x > 0 and p ≥ 1, we have where c p = (p -1) p-1 /p p ∈ [2 -p ; 1]. Denote x * 0 = 0 and 1 ≤ x * 2 ≤ · · · ≤ x * n be the nondecreasing rearrangement of the se-quence (x i ). Notice that by (23) for δ ∈ [x * j-1 ; x * j ], j = 1, 2, . . . , n, we have On the other hand, By (24) and (25) we get Similarly, denoting by y * i and z * i the nondecreasing rearrangements of the sequences Now, by (27) TV From this and from (31) we get but this (recall (19)) contradicts (30). Thus the sequence (f n ) ∞ n=1 converges in the U p ([a, b], W ) norm to f ∞ . Since the sequence (f n ) ∞ n=1 was chosen in an arbitrary way, this proves that U p ([a, b], W ) is complete.

Remark 3
It is easy to see that the space U p ([a, b], W ) equipped with the norm · TV,p, [a,b] is not separable. To see this, it suffices for two distinct vectors w 1 and w 2 from W to consider the family of functions f t : (1 A denotes the indicator function of a set A) and apply (29). However, we do not know if the subspace of continuous functions in U p ([a, b], R) is separable.

Remark 4
From the triangle inequality for · p-TV, [a,b] it follows that it is an subadditive functional of the interval, that is, for any p ≥ 1, f : To see this, it suffices to consider the decomposition f (t) = f 1 (t) + f 2 (t), f 1 (t) = 1 [a,d] (t)f (t) + 1 (d,b] (t)f (d), f 2 (t) = 1 (d,b] (t)f (t) -1 (d,b] (t)f (d). We naturally have However, superadditivity, as a function of the interval, holding for · In this subsection, we prove the following result. Proof Let L be the least positive integer such that sup t∈ [a,b] f (t) W ≤ 2 L . Consider the partition π = {a ≤ t 0 < t 1 < · · · < t n ≤ b} such that f (t i ) = f (t i-1 ) for i = 1, 2, . . . , n and for j = 0, 1, . . . , define and δ(j) := 2 L-j-1 . Naturally, for i ∈ I j , By the first assumption in (32) we have that, for all j = 0, 1, . . . , One more application of Theorem 3 is the following. Assume that y : [a, b] → E is a solution of the equation of the form where x ∈ U q ([a, b], E) is continuous, and F(·, y(·)) ∈ U p ([a, b], L(E, V )) for some p > 1 and q > 1 such that p -1 + q -1 > 1; from this by Theorem 3 we will obtain that y ∈ U q ([a, b], E).
In our case we have no longer the supperadditivity property of the functional · p p-TV, [a,b] as the function of interval (see Remark 4), and hence the method of the proof of Theorem 3 will be different from those of related estimates in [15]. It will be similar to the proof of Corollary 1. We need the following lemma.