scholarly journals Martingale decomposition of an L2 space with nonlinear stochastic integrals

2019 ◽  
Vol 56 (4) ◽  
pp. 1231-1243
Author(s):  
Clarence Simard
Keyword(s):  
Integral Representation ◽  
General Result ◽  
Optimization Problem ◽  
Stochastic Integrals ◽  
Polynomial Functions ◽  
Spatial Parameter ◽  
The Family

AbstractThis paper generalizes the Kunita–Watanabe decomposition of an $L^2$ space. The generalization comes from using nonlinear stochastic integrals where the integrator is a family of continuous martingales bounded in $L^2$ . This result is also the solution of an optimization problem in $L^2$ . First, martingales are assumed to be stochastic integrals. Then, to get the general result, it is shown that the regularity of the family of martingales with respect to its spatial parameter is inherited by the integrands in the integral representation of the martingales. Finally, an example showing how the results of this paper, with the Clark–Ocone formula, can be applied to polynomial functions of Brownian integrals.

scholarly journals On the integral representation of the endomorphisms of Mikuśinski's operator field

1971 ◽  
Vol 17 (4) ◽  
pp. 351-367 ◽  
Author(s):  
András Bleyer
Keyword(s):  
Integral Representation ◽  
Polynomial Functions ◽  
Piecewise Polynomial ◽  
Operator Field ◽  
Piecewise Polynomial Functions

We proved in (1) that every continuous endomorphism can be generated on a subring of the field M. More precisely, the ring H of piecewise polynomial functions has the property that every isomorphism from H into M, continuous in the sequential topology of H, can be extended to a continuous endomorphism of M where the notion of continuity in M is the usual sequential one.

scholarly journals Some Difference Equations for Srivastava’s λ-Generalized Hurwitz–Lerch Zeta Functions with Applications

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 311 ◽  
Author(s):  
Asifa Tassaddiq
Keyword(s):  
Integral Representation ◽  
Difference Equations ◽  
Zeta Functions ◽  
Numerical Computations ◽  
Direct Evaluation ◽  
Mellin Transforms ◽  
The Family

In this article, we establish some new difference equations for the family of λ-generalized Hurwitz–Lerch zeta functions. These difference equations proved worthwhile to study these newly defined functions in terms of simpler functions. Several authors investigated such functions and their analytic properties, but no work has been reported for an estimation of their values. We perform some numerical computations to evaluate these functions for different values of the involved parameters. It is shown that the direct evaluation of involved integrals is not possible for the large values of parameter s; nevertheless, using our new difference equations, we can evaluate these functions for the large values of s. It is worth mentioning that for the small values of this parameter, our results are 100% accurate with the directly computed results using their integral representation. Difference equations so obtained are also useful for the computation of some new integrals of products of λ-generalized Hurwitz–Lerch zeta functions and verified to be consistent with the existing results. A derivative property of Mellin transforms proved fundamental to present this investigation.

scholarly journals The functional Itō formula under the family of continuous semimartingale measures

2016 ◽  
Vol 16 (04) ◽  
pp. 1650010 ◽  
Author(s):  
Harald Oberhauser
Keyword(s):  
Continuous Dependence ◽  
Quadratic Variation ◽  
Stochastic Integrals ◽  
Itô’S Formula ◽  
Itô Formula ◽  
Ito Formula ◽  
Underlying Process ◽  
Continuous Semimartingale ◽  
The Family ◽  
Nonlinear Functionals

Dupire [16] introduced a notion of smoothness for functionals of paths and arrived at a generalization of Itō’s formula that applies to functionals with a continuous dependence on the trajectories of the underlying process. In this paper, we study nonlinear functionals that do not have such continuity. By revisiting old work of Bichteler and Karandikar we show that one can construct pathwise versions of complex functionals like the quadratic variation, stochastic integrals or Itō processes that are still regular enough such that a functional Itō-formula applies.

Quasi-free stochastic integral representation theorems over the CCR

1988 ◽  
Vol 104 (2) ◽  
pp. 383-398 ◽  
Author(s):  
Ivan F. Wilde
Keyword(s):  
Hilbert Space ◽  
Integral Representation ◽  
Stochastic Integral ◽  
Stochastic Integrals ◽  
Representation Theorems ◽  
Martingale Representation ◽  
Stochastic Integral Representation ◽  
Vector Valued ◽  
General Vector

AbstractIt is shown that each vector in the Hilbert space of certain quasi-free representations of the CCR can be written uniquely in terms of quantum stochastic integrals. As a consequence, we obtain general vector-valued and operator-valued boson quantum martingale representation theorems.

scholarly journals Optimizing phylogenetic supertrees using answer set programming

2015 ◽  
Vol 15 (4-5) ◽  
pp. 604-619 ◽  
Author(s):  
LAURA KOPONEN ◽  
EMILIA OIKARINEN ◽  
TOMI JANHUNEN ◽  
LAURA SÄILÄ
Keyword(s):  
Phylogenetic Trees ◽  
Optimization Problem ◽  
Matrix Representation ◽  
Answer Set Programming ◽  
Heuristic Methods ◽  
Construction Problem ◽  
Single Tree ◽  
Conflicting Information ◽  
The Family ◽  
Answer Set

AbstractThe supertree construction problem is about combining several phylogenetic trees with possibly conflicting information into a single tree that has all the leaves of the source trees as its leaves and the relationships between the leaves are as consistent with the source trees as possible. This leads to an optimization problem that is computationally challenging and typically heuristic methods, such as matrix representation with parsimony (MRP), are used. In this paper we consider the use of answer set programming to solve the supertree construction problem in terms of two alternative encodings. The first is based on an existing encoding of trees using substructures known as quartets, while the other novel encoding captures the relationships present in trees through direct projections. We use these encodings to compute a genus-level supertree for the family of cats (Felidae). Furthermore, we compare our results to recent supertrees obtained by the MRP method.

On the Darboux Transformation. I

1995 ◽  
Vol 2 (3) ◽  
pp. 237-240
Author(s):  
Veronika Chrastinová
Keyword(s):  
Darboux Transformation ◽  
General Result ◽  
The Family ◽  
Sturm Liouville

Abstract Automorphisms of the family of all Sturm-Liouville equations y″ = qy are investigated. The classical Darboux transformation arises as a particular case of a general result.

Distortion inequality for a Markov operator generated by a randomly perturbed family of Markov Maps in ℝ d

2021 ◽  
Vol 11 (1) ◽  
pp. 225-242
Author(s):  
Peter Bugiel ◽  
Stanisław Wędrychowicz ◽  
Beata Rzepka
Keyword(s):  
Probability Distribution ◽  
General Result ◽  
Probability Space ◽  
Asymptotic Properties ◽  
Markov Operator ◽  
Random Vectors ◽  
The Family ◽  
Random Elements ◽  
Common Probability ◽  
Additional Assumptions

Abstract Asymptotic properties of the sequences (a) { P j } j = 1 ∞ $\{P^{j}\}_{j=1}^{\infty}$ and (b) { j − 1 ∑ i = 0 j − 1 P i } j = 1 ∞ $\{ j^{-1} \sum _{i=0}^{j-1} P^{i}\}_{j=1}^{\infty}$ are studied for g ∈ G = {f ∈ L 1(I) : f ≥ 0 and ‖f ‖ = 1}, where P : L 1(I) → L 1(I) is a Markov operator defined by P f := ∫ P y f d p ( y ) $Pf:= \int P_{y}f\, dp(y) $ for f ∈ L 1; {Py } y∈Y is the family of the Frobenius-Perron operators associated with a family {φy } y∈Y of nonsingular Markov maps defined on a subset I ⊆ ℝ d ; and the index y runs over a probability space (Y, Σ(Y), p). Asymptotic properties of the sequences (a) and (b), of the Markov operator P, are closely connected with the asymptotic properties of the sequence of random vectors x j = φ ξ j ( x j − 1 ) $x_{j}=\varphi_{\xi_{j}}(x_{j-1})$ for j = 1,2, . . .,where { ξ j } j = 1 ∞ $\{\xi_{j}\}_{j=1}^{\infty}$ is a sequence of Y-valued independent random elements with common probability distribution p. An operator-theoretic analogue of Rényi’s Condition is introduced for the family {Py } y∈Y of the Frobenius-Perron operators. It is proved that under some additional assumptions this condition implies the L 1- convergence of the sequences (a) and (b) to a unique g 0 ∈ G. The general result is applied to some families {φy } y∈Y of smooth Markov maps in ℝ d .

scholarly journals Optimization Problem in Series Systems with Random Number of Components from the Family of Power Series Distributions

2022 ◽  
Vol 15 (2) ◽  
pp. 481-504
Author(s):  
Motahare ZaeamZadeh ◽  
Jafar Ahmadi ◽  
Bahareh Khatib Astaneh ◽  
◽  
◽  
...  
Keyword(s):  
Power Series ◽  
Optimization Problem ◽  
Random Number ◽  
Number Of Components ◽  
The Family ◽  
In Series ◽  
Power Series Distributions ◽  
Series Systems

Integral representation of functionals defined on curves of W1,p

1998 ◽  
Vol 128 (2) ◽  
pp. 193-217 ◽  
Author(s):  
Micol Amar ◽  
Giovanni Bellettini ◽  
Sergio Venturini
Keyword(s):  
Integral Representation ◽  
Representation Theorem ◽  
Lower Semicontinuous ◽  
Borel Function ◽  
Open Interval ◽  
Growth Conditions ◽  
The Family ◽  
Representation Result ◽  
Image Position

Let I ⊂ ℝ be a bounded open interval, (I) be the family of all open subintervals of I and let p > 1. The aim of this paper is to give an integral representation result for abstract functionals F: W1,p(I;ℝn) × (I) → [0, + ∞) which are lower semicontinuous and satisfy suitable properties. In particular, we prove an integral representation theorem for the Г-limit of a sequence {Fh}h, of functionals of the formwhere each fh is a Borel function satisfying proper growth conditions.

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