Continuous and Discrete Approximations

2014 ◽  
pp. 249-280
Author(s):  
Tom Lyche ◽  
Jean-Louis Merrien
Keyword(s):  
Discrete Approximations

scholarly journals Non-local Solvable Birth–Death Processes

2021 ◽  
Author(s):  
Giacomo Ascione ◽  
Nikolai Leonenko ◽  
Enrica Pirozzi
Keyword(s):  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Limit Distribution ◽  
Spectral Decomposition ◽  
Strong Solutions ◽  
Stochastic Representation ◽  
Discrete Approximations ◽  
Local Difference ◽  
Non Local ◽  
Birth Death

AbstractIn this paper, we study strong solutions of some non-local difference–differential equations linked to a class of birth–death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed birth–death processes and study their invariant and their limit distribution. Finally, we describe the correlation structure of the aforementioned time-changed birth–death processes.

scholarly journals Dynkin's Games and Israeli Options

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Yuri Kifer
Keyword(s):  
Transaction Costs ◽  
Error Estimates ◽  
Optimal Stopping ◽  
Derivative Securities ◽  
Discrete Approximations ◽  
Game Options ◽  
Related Derivative ◽  
Stopping Games

We start by briefly surveying a research on optimal stopping games since their introduction by Dynkin more than 40 years ago. Recent renewed interest to Dynkin’s games is due, in particular, to the study of Israeli (game) options introduced in 2000. We discuss the work on these options and related derivative securities for the last decade. Among various results on game options we consider error estimates for their discrete approximations, swing game options, game options in markets with transaction costs, and other questions.

scholarly journals The combinatorics of discrete time-trees: theory and open problems

2016 ◽  
Author(s):  
Alex Gavryushkin ◽  
Chris Whidden ◽  
Frederick A Matsen
Keyword(s):  
Shortest Paths ◽  
Point Of View ◽  
Upper And Lower Bounds ◽  
Evolutionary Divergence ◽  
Discrete Approximations ◽  
Open Problems ◽  
Graph Theoretic ◽  
Tree Graphs ◽  
Object Of Study ◽  
Rooted Phylogenetic Tree

ABSTRACTA time-tree is a rooted phylogenetic tree such that all internal nodes are equipped with absolute divergence dates and all leaf nodes are equipped with sampling dates. Such time-trees have become a central object of study in phylogenetics but little is known about the parameter space of such objects. Here we introduce and study a hierarchy of discrete approximations of the space of time-trees from the graph-theoretic and algorithmic point of view. One of the basic and widely used phylogenetic graphs, the NNI graph, is the roughest approximation and bottom level of our hierarchy. More refined approximations discretize the relative timing of evolutionary divergence and sampling dates. We study basic graph-theoretic questions for these graphs, including the size of neighborhoods, diameter upper and lower bounds, and the problem of finding shortest paths. We settle many of these questions by extending the concept of graph grammars introduced by Sleator, Tarjan, and Thurston to our graphs. Although time values greatly increase the number of possible trees, we show that 1-neighborhood sizes remain linear, allowing for efficient local exploration and construction of these graphs. We also obtain upper bounds on the r-neighborhood sizes of these graphs, including a smaller bound than was previously known for NNI.Our results open up a number of possible directions for theoretical investigation of graph-theoretic and algorithmic properties of the time-tree graphs. We discuss the directions that are most valuable for phylogenetic applications and give a list of prominent open problems for those applications. In particular, we conjecture that the split theorem applies to shortest paths in time-tree graphs, a property not shared in the general NNI case.

scholarly journals Microtubules: A network for solitary waves

2017 ◽  
Vol 82 (5) ◽  
pp. 469-481 ◽  
Author(s):  
Slobodan Zdravkovic
Keyword(s):  
Nonlinear Dynamics ◽  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Solitary Waves ◽  
Discrete Approximations ◽  
Mathematical Methods ◽  
In Cells

In the present paper we deal with nonlinear dynamics of microtubules. The structure and role of microtubules in cells are explained as well as one of models explaining their dynamics. Solutions of the crucial nonlinear differential equation depend on used mathematical methods. Two commonly used procedures, continuum and semi-discrete approximations, are explained. These solutions are solitary waves usually called as kink solitons, breathers and bell-type solitons.

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