Deviation inequalities for continuous martingales

In this paper we define a new type of quadratic variation for cylindrical continuous local martingales on an infinite dimensional spaces. It is shown that a large class of cylindrical continuous local martingales has such a quadratic variation. For this new class of cylindrical continuous local martingales we develop a stochastic integration theory for operator valued processes under the condition that the range space is a UMD Banach space. We obtain two-sided estimates for the stochastic integral in terms of the $\gamma $-norm. In the scalar or Hilbert case this reduces to the Burkholder-Davis-Gundy inequalities. An application to a class of stochastic evolution equations is given at the end of the paper.

We present a new test for the “continuous martingale hypothesis”. That is, a test for the hypothesis that observed data are from a process which is a continuous local martingale. The basis of the test is an embedded random walk at first passage times, obtained from the well-known representation of a continuous local martingale as a continuous time-change of Brownian motion. With a variety of simulated diffusion processes our new test shows higher power than existing tests using either the crossing tree or the quadratic variation, including the situation where non-negligible drift is present. The power of the test in the presence of jumps is also explored with a variety of simulated jump diffusion processes. The test is also applied to two sequences of high-frequency foreign exchange trade-by-trade data. In both cases the continuous martingale hypothesis is rejected at times less than hourly and we identify significant dependence in price movements at these small scales.

The following results are obtained for the metric of a sign sensitive weight: necessary and sufficient conditions imposed on the weight in order to ensure the fulfillment of the complete analogue of Jackson's theorem on the estimate of the best approximation of an arbitrary continuous function by means of its modulus of continuity; the analogue of Bernstein's inequality on the estimate of the derivative of a trigonometric polynomial; the analogue of Stechkin's theorem on the connection between the modulus of continuity of a function and the rate of its approximation by polynomials; the analogue of Dolzhenko's inequality on the estimate of the variation of a rational function; and the analogue of Dolzhenko's theorem on the estimate of the variation of a function by means of its best rational approximation.

Covariation de convolution de martingales

We propose a model of a passive nerve cylinder undergoing random stimulus along its length. It is shown that this model is approximated by the solution of a stochastic partial differential equation. Numerous properties of the sample paths are derived, such as their modulus of continuity, quadratic and quartic variation, and it is shown that the solution exhibits the phenomenon of flicker noise. The first-passage problem is studied, and it is shown to be connected with a first-hitting time for an infinite-dimensional diffusion.

We consider cadlag local martingales M with initial value zero and jumps larger than a for some a larger than or equal to -1, and prove Novikov-type criteria for an exponential local martingale to be a uniformly integrable martingale. We obtain criteria using both the quadratic variation and the predictable quadratic variation. We prove optimality of the coefficients in the criteria. As a corollary, we obtain a verbatim extension of the classical Novikov criterion for continuous local martingales to the case of local martingales with initial value zero and nonnegative jumps.

The well-known treatment of the path integral for the Coulomb potential, by means of the Kustaanheimo–Stiefel transformation and a time transformation, is made more transparent by reducing the problem to the equality of two measures on the space of paths. For this equality two proofs are given: an elementary computational one and a short one recurring to general features of stochastic processes. It is shown that the time transformation is a special case of the well-known time change for a continuous local martingale, by which the process is changed to a Brownian motion and which is determined by its quadratic variation.

In this article, we consider a bivariate Chlodowsky type Szász–Durrmeyer operators on weighted spaces. We obtain the rate of approximation in connection with the partial and complete modulus of continuity and also for the elements of the Lipschitz type class. Moreover, we examine the degree of convergence with regard to the weighted modulus of continuity and Peetre’s K-functional. Further, we construct the associated GBS type of these operators and estimate the degree of approximation using the mixed modulus of continuity and a class of the Lipschitz of Bögel type continuous functions. Finally, with the help of Maple software, we present the comparisons of the convergence of the bivariate Chlodowsky type Szász–Durrmeyer operators and associated GBS type operators to certain functions with some graphs and error estimation tables.

Asymptotic equalities are obtained for upper bounds of the deviations of Fourier sums in the classes of convolutions of Poisson kernels and continuous functions with moduli of continuity not exceeding fixed majorants.

This note explores the common core of constructive, intuitionistic, recursive and classical analysis from an axiomatic standpoint. In addition to clarifying the relation between Kleene’s and Troelstra’s minimal formal theories of numbers and number-theoretic sequences, we propose some modified choice principles and other function existence axioms which may be of use in reverse constructive analysis. Specifically, we consider the function comprehension principles assumed by the two minimal theories EL and M, introduce an axiom schema CFd asserting that every decidable property of numbers has a characteristic function, and use it to describe a precise relationship between the minimal theories. We show that the axiom schema AC00 of countable choice can be decomposed into a monotone choice schema AC 00 m (which guarantees that every Cauchy sequence has a modulus) and a bounded choice schema BC00. We relate various (classically correct) axiom schemas of continuous choice to versions of the bar and fan theorems, suggest a constructive choice schema AC1/2,0 (which incidentally guarantees that every continuous function has a modulus of continuity), and observe a constructive equivalence between restricted versions of the fan theorem and correspondingly restricted bounding axioms $${AB_{1/2,0}^{2^{\mathbb{N}}}}$$ . We also introduce a version WKL!! of Weak König’s Lemma with uniqueness which is intermediate in strength between WKL and the decidable fan theorem FTd.

In this note we consider a linear and positive compound approximation operator of D.D. Stancu type depending of several parameters; we give the expressions of this operator on the test functions, the conditions under which this operator converges to a given continuous function, an estimate of the order of approximation using the moduli of continuity and an integral representation of the remainder. Also, by using Stancu's method we find an expression for the remainder using divided differences of second order for a special case of this operator.

We prove a statement on exact inequalities between the deviations of functions from their linear methods (in the metric of L2) with multipliers defined by a continuous function and majorants determined as the scalar product of the squared modulus of continuity (of order r) in L2 for the lth derivative of the function and a certain weight function θ. We obtain several corollaries of the general theorem.

We investigate some new inequalities by estimating the rate of convergence by means of the complete modulus of continuity and a class of Lipschitz functions for the bivariate Bernstein-type operator including Bézier basis and present an example including numerical results comparing its rate of convergence. Moreover, we introduce its GBS (Generalized Boolean Sum) operator and obtain its rate of convergence with the help of the mixed modulus of continuity and Lipschitz class of Bögel continuous functions with exemplifying numerical results. Our research will demonstrate that the GBS operator possesses at least better numerical results than the bivariate Bernstein-type operator. All mentioned results point out the novelty of this study.

Let \(\nu \) be a nondecreasing concave sequence of positive real numbers and \(1\le p<\infty \). In this note, we introduce the notion of modulus of p-variation for a function of a real variable, and show that it serves in at least two important problems, namely, the uniform convergence of Fourier series and computation of certain K-functionals. Using this new tool, we first define a Banach space, denoted \(V_p[\nu ]\), that is a natural unification of the Wiener class \(BV_p\) and the Chanturiya class \(V[\nu ]\). Then we prove that \(V_p[\nu ]\) satisfies a Helly-type selection principle which enables us to characterize continuous functions in \(V_p[\nu ]\) in terms of their Fejér means. We also prove that a certain K-functional for the couple \((C,BV_p)\) can be expressed in terms of the modulus of p-variation, where C denotes the space of continuous functions. Next, we obtain equivalent optimal conditions for the uniform convergence of the Fourier series of all functions in each of the classes \(C\cap V_p[\nu ]\) and \(H^\omega \cap V_p[\nu ]\), where \(\omega \) is a modulus of continuity and \(H^\omega \) denotes its associated Lipschitz class. Finally, we establish sharp embeddings into \(V_p[\nu ]\) of various spaces of functions of generalized bounded variation. As a by-product of these latter results, we infer embedding results for certain symmetric sequence spaces.

A new network with super-approximation power is introduced. This network is built with Floor (⌊x⌋) or ReLU (max{0,x}) activation function in each neuron; hence, we call such networks Floor-ReLU networks. For any hyperparameters N∈N+ and L∈N+, we show that Floor-ReLU networks with width max{d,5N+13} and depth 64dL+3 can uniformly approximate a Hölder function f on [0,1]d with an approximation error 3λdα/2N-αL, where α∈(0,1] and λ are the Hölder order and constant, respectively. More generally for an arbitrary continuous function f on [0,1]d with a modulus of continuity ωf(·), the constructive approximation rate is ωf(dN-L)+2ωf(d)N-L. As a consequence, this new class of networks overcomes the curse of dimensionality in approximation power when the variation of ωf(r) as r→0 is moderate (e.g., ωf(r)≲rα for Hölder continuous functions), since the major term to be considered in our approximation rate is essentially d times a function of N and L independent of d within the modulus of continuity.