scholarly journals The existence of a multiple spider martingale in the natural filtration of a certain diffusion in the plane

1999 ◽  
pp. 277-290 ◽  
Author(s):  
Shinzo Watanabe
Keyword(s):  
Natural Filtration

scholarly journals Path homology theory of edge-colored graphs

2021 ◽  
Vol 19 (1) ◽  
pp. 706-723
Author(s):  
Yuri V. Muranov ◽  
Anna Szczepkowska
Keyword(s):  
Spectral Sequence ◽  
Edge Coloring ◽  
Homology Theory ◽  
Homotopy Category ◽  
Homology Groups ◽  
Colored Graphs ◽  
Natural Filtration ◽  
Definition Of ◽  
Exact Sequences

Abstract In this paper, we introduce the category and the homotopy category of edge-colored digraphs and construct the functorial homology theory on the foundation of the path homology theory provided by Grigoryan, Muranov, and Shing-Tung Yau. We give the construction of the path homology theory for edge-colored graphs that follows immediately from the consideration of natural functor from the category of graphs to the subcategory of symmetrical digraphs. We describe the natural filtration of path homology groups of any digraph equipped with edge coloring, provide the definition of the corresponding spectral sequence, and obtain commutative diagrams and braids of exact sequences.

Stopping sets: Gamma-type results and hitting properties

1999 ◽  
Vol 31 (2) ◽  
pp. 355-366 ◽  
Author(s):  
Sergei Zuyev
Keyword(s):  
Point Process ◽  
Sampling Theorem ◽  
Random Sets ◽  
Scaling Properties ◽  
Stopping Sets ◽  
Stopping Set ◽  
Alternative Description ◽  
Natural Filtration ◽  
Random Figure ◽  
Optional Sampling Theorem

Recently in the paper by Møller and Zuyev (1996), the following Gamma-type result was established. Given n points of a homogeneous Poisson process defining a random figure, its volume is Γ(n,λ) distributed, where λ is the intensity of the process. In this paper we give an alternative description of the class of random sets for which the Gamma-type results hold. We show that it corresponds to the class of stopping sets with respect to the natural filtration of the point process with certain scaling properties. The proof uses the martingale technique for directed processes, in particular, an analogue of Doob's optional sampling theorem proved in Kurtz (1980). As well as being compact, this approach provides a new insight into the nature of geometrical objects constructed with respect to a Poisson point process. We show, in particular, that in this framework the probability that a point is covered by a stopping set does not depend on whether it is a point of the process or not.

scholarly journals A General ‘Bang-Bang’ Principle for Predicting the Maximum of a Random Walk

2010 ◽  
Vol 47 (04) ◽  
pp. 1072-1083 ◽  
Author(s):  
Pieter Allaart
Keyword(s):  
Random Walk ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Stopping Times ◽  
Optimal Prediction ◽  
Prediction Problem ◽  
Brownian Motion With Drift ◽  
Natural Filtration ◽  
Mathematical Justification ◽  
Bernoulli Random Walk

Let (B t )0≤t≤T be either a Bernoulli random walk or a Brownian motion with drift, and let M t := max{B s: 0 ≤ s ≤ t}, 0 ≤ t ≤ T. In this paper we solve the general optimal prediction problem sup0≤τ≤T E[f(M T − B τ], where the supremum is over all stopping times τ adapted to the natural filtration of (B t ) and f is a nonincreasing convex function. The optimal stopping time τ* is shown to be of ‘bang-bang’ type: τ* ≡ 0 if the drift of the underlying process (B t ) is negative and τ* ≡ T if the drift is positive. This result generalizes recent findings of Toit and Peskir (2009) and Yam, Yung and Zhou (2009), and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good stocks as long as possible.

scholarly journals A General ‘Bang-Bang’ Principle for Predicting the Maximum of a Random Walk

2010 ◽  
Vol 47 (4) ◽  
pp. 1072-1083 ◽  
Author(s):  
Pieter Allaart
Keyword(s):  
Random Walk ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Stopping Times ◽  
Optimal Prediction ◽  
Prediction Problem ◽  
Brownian Motion With Drift ◽  
Natural Filtration ◽  
Mathematical Justification ◽  
Bernoulli Random Walk

Let (Bt)0≤t≤T be either a Bernoulli random walk or a Brownian motion with drift, and let Mt := max{Bs: 0 ≤ s ≤ t}, 0 ≤ t ≤ T. In this paper we solve the general optimal prediction problem sup0≤τ≤TE[f(MT − Bτ], where the supremum is over all stopping times τ adapted to the natural filtration of (Bt) and f is a nonincreasing convex function. The optimal stopping time τ* is shown to be of ‘bang-bang’ type: τ* ≡ 0 if the drift of the underlying process (Bt) is negative and τ* ≡ T if the drift is positive. This result generalizes recent findings of Toit and Peskir (2009) and Yam, Yung and Zhou (2009), and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good stocks as long as possible.

THE (2, ∞)-SKEIN MODULE OF WHITEHEAD MANIFOLDS

1995 ◽  
Vol 04 (03) ◽  
pp. 411-427 ◽  
Author(s):  
JIM HOSTE ◽  
JÓZEF H. PRZYTYCKI
Keyword(s):  
Jones Polynomial ◽  
Skein Modules ◽  
Skein Module ◽  
Natural Filtration ◽  
Stark Contrast ◽  
Torsion Free

To any orientable 3-manifold one can associate a module, called the (2, ∞)-skein module, which is essentially a generalization of the Jones polynomial of links in S3. For an uncountable collection of open contractible 3-manifolds, each constructed in a fashion similar to the classic Whitehead manifold, we prove that their (2, ∞)-skein modules are infinitely generated, torsion free, but not free. These examples stand in stark contrast to [Formula: see text], whose (2, ∞)-skein module is free on one generator. To each of these manifolds we associate a subgroup G of the rationals which may be interpreted via wrapping numbers. We show that the skein module of M has a natural filtration by modules indexed by G. For the specific case of the Whitehead manifold, we describe its (2, ∞)-skein module and associated filtration in greater detail.

scholarly journals Bridges with Random Length: Gamma Case

2019 ◽  
Vol 33 (2) ◽  
pp. 931-953
Author(s):  
Mohamed Erraoui ◽  
Astrid Hilbert ◽  
Mohammed Louriki
Keyword(s):  
Markov Process ◽  
Jump Process ◽  
Canonical Decomposition ◽  
Random Interval ◽  
Basic Properties ◽  
Natural Filtration ◽  
The Given

Abstract In this paper, we generalize the concept of gamma bridge in the sense that the length will be random, that is, the time to reach the given level is random. The main objective of this paper is to show that certain basic properties of gamma bridges with deterministic length stay true also for gamma bridges with random length. We show that the gamma bridge with random length is a pure jump process and that its jumping times are countable and dense in the random interval bounded by 0 and the random length. Moreover, we prove that this process is a Markov process with respect to its completed natural filtration as well as with respect to the usual augmentation of this filtration, which leads us to conclude that its completed natural filtration is right continuous. Finally, we give its canonical decomposition with respect to the usual augmentation of its natural filtration.

scholarly journals NATURAL FILTRATION EQUATIONS. FIASCO “OF DARCY'S LAW”

2018 ◽  
pp. 54-70
Author(s):  
Jakupov K. B.
Keyword(s):  
Continuity Equation ◽  
Darcy’S Law ◽  
Power Laws ◽  
Darcy's Law ◽  
Filtration Theory ◽  
Friction Law ◽  
Natural Filtration ◽  
Continuum Dynamics

The theory of natural filtration equations is given. The naturalness of the new filtration equations is that they are the exact consequences of the fundamental laws of physics, directly take into account the density and porosity of the soil, the viscosity and density of the filtration fluid, drainage, the influence of gravity, etc.the falsity of the traditional continuity equation in the filtration theory is Established. New filtration equations are derived from the equation of continuum dynamics in stresses, including the density and viscosity of the liquid and the porosity of the soil.Inadequacy of the modeling filter equations with the friction law of Newton. The efficiency of simulation of filtration by Jakupova equations based on the power laws of friction with odd exponents is numerically confirmed, with the use of which the calculations of filtration in the well, drainage under the influence of gravity, displacement of oil by water from the underground area through two symmetrically located pits are carried out.

Aqueous Suspensions of Cellulose Oligomer Nanoribbons for Growth and Natural Filtration-Based Separation of Cancer Spheroids

Langmuir ◽  
2020 ◽  
Vol 36 (46) ◽  
pp. 13890-13898
Author(s):  
Takeshi Serizawa ◽  
Tohru Maeda ◽  
Saeko Yamaguchi ◽  
Toshiki Sawada
Keyword(s):  
Aqueous Suspensions ◽  
Natural Filtration ◽  
Cancer Spheroids

Type-two invariants for knots in the solid torus

2016 ◽  
Vol 25 (08) ◽  
pp. 1650051 ◽  
Author(s):  
Khaled Bataineh
Keyword(s):  
Linear Combination ◽  
Solid Torus ◽  
Winding Number ◽  
Vassiliev Invariants ◽  
Natural Filtration ◽  
Gauss Diagram ◽  
Singular Knots

We introduce a natural filtration in the space of knots and singular knots in the solid torus, and start the study of the type-two Vassiliev invariants with respect to this filtration. The main result of the work states that any such invariant within the second term of this filtration in the space of knots with zero winding number is a linear combination of seven explicitly described Gauss diagram invariants. This introduces a basis (and a universal invariant) for the type-two Vassiliev invariants for knots with zero winding number. Then we formalize the problem of exploring the set of all type-two invariants for knots with zero winding number.

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