Covariation de convolution de martingales

A set of some predictable fuzzy stochastic processes is a complete metric space for some metric. A set of some square integrable fuzzy martingales is a complete metric space for some metric and the set of all almost surely continuous square integrable martingales is its closed subset.

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.

Deviation inequalities for continuous martingales

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.

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.

This paper supplements an earlier one by the authors which constructed the Dedekind completion of the ring of continuous real functions on an arbitrary frame L in terms of partial continuous real functions on L. In the present paper, we provide three alternative views of it, in terms of (i) normal semicontinuous real functions on L, (ii) the Booleanization of L (in the case of bounded real functions) and the Gleason cover of L (in the general case), and (iii) Hausdorff continuous partial real functions on L. The first is the normal completion and extends Dilworth’s classical construction to the pointfree setting. The second shows that in the bounded case, the Dedekind completion is isomorphic to the lattice of bounded continuous real functions on the Booleanization of L, and that in the non-bounded case, it is isomorphic to the lattice of continuous real functions on the Gleason cover of L. Finally, the third is the pointfree version of Anguelov’s approach in terms of interval-valued functions. Two new classes of frames, cb-frames and weak cb-frames, emerge naturally in the first two representations. We show that they are conservative generalizations of their classical counterparts.

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 chapter, we consider studies involving cadlag processes. We pay particular attention to semi-martingales with respect to a given filtration; these processes will always be taken with cadlag paths. We present the definition of stochastic integrals with respect to a square integrable martingale, and we extend the definition to stochastic integrals with respect to a local martingale. Then, we introduce semi-martingales, quadratic covariation processes for semi-martingales and some general versions of Ito’s formula and Girsanov’s theorem. We give necessary and sufficient conditions for the existence of an equivalent martingale measure. We end the chapter with a brief survey of valuation in an incomplete market.

In computable analysis, sequences of rational numbers which effectively converge to a real number x are used as the (rho-) names of x. A real number x is computable if it has a computable name, and a real function f is computable if there is a Turing machine M which computes f in the sense that, M accepts any rho-name of x as input and outputs a rho-name of f(x) for any x in the domain of f. By weakening the effectiveness requirement of the convergence and classifying the converging speeds of rational sequences, several interesting classes of real numbers of weak computability have been introduced in literature, e.g., in addition to the class of computable real numbers (EC), we have the classes of semi-computable (SC), weakly computable (WC), divergence bounded computable (DBC) and computably approximable real numbers (CA). In this paper, we are interested in the weak computability of continuous real functions and try to introduce an analogous classification of weakly computable real functions. We present definitions of these functions by Turing machines as well as by sequences of rational polygons and prove these two definitions are not equivalent. Furthermore, we explore the properties of these functions, and among others, show their closure properties under arithmetic operations and composition.

Extended real functions in pointfree topology

An example is given of a real connectivity function which is not the composition of any finite collection of almost continuous functions. We also investigate conditions under which the composition of two real Darboux functions can be continuous. In [6] I asked if every Darboux function from < to < is the composition of two almost continuous functions. In the present note it is shown that the answer is “no”. In fact, there exits a connectivity function which cannot be written as the composition of finitely many almost continuous functions. This example is not particularly difficult. I think we have overlooked this example until now because no one expected it to exist. The composition of almost continuous functions can be very nasty. Our function which is not such a composition is as well behaved as a Darboux function can be and not be almost continuous. Natkaniec [7] has shown that if f is Darboux and nasty–that is, f−1(x) is c-dense for every x, then f is the composition of two almost continuous functions. To look at the other extreme, we need a Darboux function which is not almost continuous but as nice as possible. Since a Darboux function of Baire class 1 is almost continuous [1], the function we want must be totally discontinuous on some perfect set. Jones and Thomas [5] give an example of a function which is connectivity, continuous on the complement of the Cantor set, but not almost continuous. Our example is a simple modification of the Jones and Thomas example. A function f : < → < is almost continuous if, given a closed set K ⊂ < such that graph(f) ∩ K = ∅, there exists a continuous function g : < → <

Riemann zeta is defined as a function of a complex variable that analytically continues the sum of the Dirichlet series, when the real part is greater than unity. In this paper, the Riemann zeta associated with the finite energy possessed by a 2mm radius, free falling water droplet, crashing into a plane is considered. A modified zeta function is proposed which is incorporated to the spherical coordinates and real analysis has been performed. Through real analytic continuation, the single point of contact of the drop at the instant of touching the plane is analyzed. The zeta function is extracted at the point of destruction of the drop, where it defines a unique real function. A special property is assumed for some continuous functions, where the function’s first derivative and first integral combine together to a nullity at all points. Approximate reverse synthesis of such a function resulted in a special waveform named the dying-surge. Extending the proposed concept to general continuous real functions resulted in the synthesis of the corresponding function’s Dying-surge model. The Riemann zeta function associated with the water droplet can also be modeled as a dying–surge. The Dying- surge model corresponds to an electrical squeezing or compression of a waveform, which was originally defined over infinite arguments, squeezed to a finite number of values for arguments placed very close together with defined final and penultimate values. Synthesized results using simulation software are also presented, along with the analysis. The presence of surges in electrical circuits will correspond to electrical compression of some unknown continuous, real current or voltage function and the method can be used to estimate the original unknown function.

On the Kolmogorov complexity of continuous real functions

This paper introduces the frame of partially defined real numbers and the lattice-ordered ring of partial real functions on a frame. This is then used to construct the order completion of rings of pointfree continuous real functions. The bounded and integer-valued cases are also analysed. The application of this pointfree approach to the classical case C(X) of the ring of continuous real-valued functions on a topological space X yields a new construction for the Dedekind completion of C(X), considerably more direct and natural than the known procedure using Hausdorff continuous functions.

JOHN R. RICE(')1.1.Let two real functions/ and F be given such that/=/(x) is a continuous function for x£ [0, l] and F=F(al, a2, ■ ■ ■ , a", x) is a continuous function of n parameters and x(E [0, l].For simplicity of notation the point (a1, a2, ■ ■ ■ , a") in Euclidean w-space, En, is denoted by a.The domain of the parameters of F is denoted by P, a subset of En.The main problem in the theory of approximation of continuous functions may be stated as follows: Determine a*EP so that the deviation of the function F(a, x) from f(x) shall be minimized.Naturally one must define the deviation of F(a, x) from/(x) and different definitions lead to different theories.In this paper the deviation of F(a, x) from f(x) is taken to be maxI(=[o,i] | F(a, x) -f(x) \.All maxima and minima are taken over x£ [0, l] unless otherwise stated.F(a*, x) is said to be a best approximation to/The results of this paper remain valid if [0, l] is replaced by any other closed and bounded interval.By 1920 an elegant theory of Tchebycheff approximation had been developed for F(a, x) depending linearly on the parameters [2; 3; 7].Some of the main points of this linear theory are given in the following resume.Let T be a set of n functions 4>i(x), 4>2(x), • • • , </>"(x) continuous on [0, 1 ] and let P(a, x) = /,?_, a*'<£,(x) be called a T-polynomial.P(a, x) is said to be a nontrivial T-polynomal if 2~ll=i | a*| >0.T is said to be a Tchebycheff Set if every nontrivial polynomial has at most n-1 zeros, max | F(a, x) -/(x)| is said to alternate n times if there are « + l points 0^Xi<x2<Alternance is a basic concept in the theory of Tchebycheff approximations.The following theorem answers the principal questions of a general nature in the theory of linear Tchebycheff approximations.Theorem.Let T be a Tchebycheff set and let f(x) be an arbitrary function continuous on [0, l].Then A. f(x) possesses a best approximation, B. a necessary and sufficient condition that P(ao, x) be a best approximation to f(x) is that max | P(a0, x) -f(x) \ alternates at least n times, C. the best approximation to f(x) is unique.