• This unit will introduce an important class of stochastic processes, using the theory of martingales. (edu.au)
  • Topics include measure theoretic probability, martingales, filtration, and stopping theorems, elements of large deviations theory, Brownian motion and reflected Brownian motion, stochastic integration and Ito calculus and functional limit theorems. (freecomputerbooks.com)
  • The theory of Markov chains will guide our discussion as we cover topics such as martingales, random walks, Poisson process, birth and death processes, and Brownian motion. (williams.edu)
  • Martingales, renewal processes, and Brownian motion. (paradisegolf.net)
  • 1. At first we will introduce Martingales and Brownian Motions. (uni-ulm.de)
  • 2/ 4 · understand the tools and concepts from stochastic calculus: martingales, Itô's formula, Itô isometry, Feynman-Kac representation, change of measure (Girsanov transformation) and change of numeraire, · understand how the basic financial contracts work and how they relate to each other, e.g. (lu.se)
  • The purpose is to quickly introduce fundamental concepts of financial markets such as free of arbitrage and completeness as well as martingales and martingale measures. (lu.se)
  • S. Shreve, Stochastic Calculus for Finance II Continuous-Time Models, Springer. (lse.ac.uk)
  • Then, some properties of stochastic calculus are presented and compared to the classic calculus. (nimbios.org)
  • I was looking for a good book that explains at beginner-level the basic of stochastic calculus, probability and random variables, Itô and jump processes as well as Brownian Motion. (stackexchange.com)
  • Brownian motion or Ito calculus? (stackexchange.com)
  • The book Stochastic calculus for finance by Steven Shreve gives a good introduction to stochastic calculus applied to finance. (stackexchange.com)
  • Elementary Stochastic Calculus by Thomas Mikosch is an excellent introduction to the topic in a very compact way. (stackexchange.com)
  • Alternatively, Stochastic Calculus for Finance II: Continuous-Time Models by Steven Shreve is a more comprehensive reference which is very much oriented to applications in finance. (stackexchange.com)
  • If you're interested in learning about stochastic calculus outside of the context of quant finance (which I think is a better approach than learning about it solely in the context of finance), check out Stochastic Integration and Differential Equations by Protter. (stackexchange.com)
  • Stochastic Calculus problem with three processes? (stackexchange.com)
  • You will also learn how to use the wide range of tools required by Financial Mathematics, including stochastic calculus, partial differential equations, optimisation and statistics. (edu.au)
  • It is also the main building block for the theory of stochastic calculus (see MA482 Stochastic Analysis in term 2), and has played an important role in the development of financial mathematics. (warwick.ac.uk)
  • use the fundamental financial concepts to express relations between various financial contracts, · use the tools and concepts from stochastic calculus to price financial contracts assuming specific models for the underlying assets. (lu.se)
  • After a description of the Poisson process and related processes with independent increments as well as a brief look at Markov processes with a finite number of jumps, the author proceeds to introduce Brownian motion and to develop stochastic integrals and Itô's theory in the context of one-dimensional diffusion processes. (ams.org)
  • In another part of this paper, we discuss the use of the multi-scale entanglement renormalization ansatz (MERA), introduced in the study critical systems in quantum spin lattices, as a method for sampling integrals with respect to such multifractal processes. (harvard.edu)
  • We apply a wide-range of mathematical methods from statistical mechanics (ensemble approaches, large deviation theory), stochastic processes (SDEs, path-integrals) and machine learning. (qmul.ac.uk)
  • of a space of stochastic integrals. (123dok.com)
  • The theories behind Brownian motion, stochastic integrals, Ito-'s formula, measures changes and numeraires are presented and applied to option theory both for the stock and the interest rate markets. (lu.se)
  • The origin of the long-range memory in non-equilibrium systems is still an open problem as the phenomenon can be reproduced using models based on Markov processes. (mdpi.com)
  • A good example of Markov processes with spurious memory is a stochastic process driven by a non-linear stochastic differential equation (SDE). (mdpi.com)
  • An introduction provided the basic theory of Markov chains and stochastic differential equations. (nimbios.org)
  • Methods were presented for deriving stochastic ordinary or partial differential equations from Markov chains. (nimbios.org)
  • Some of the relationships between the master equation in Markov chain theory and the theory of stochastic differential equations were discussed. (nimbios.org)
  • Numerical methods for approximating solutions to Markov chains and stochastic differential equations were presented, including Gillespie's algorithm, Euler-Maruyama method, and Monte-Carlo simulations. (nimbios.org)
  • Applications of Markov chain models and stochastic differential equations were explored in problems associated with enzyme kinetics, viral kinetics, drug pharmacokinetics, gene switching, population genetics, birth and death processes, age-structured population growth, and competition, predation, and epidemic processes. (nimbios.org)
  • Abstract: Some basic definitions and notation for Markov chains are introduced. (nimbios.org)
  • Abstract: Some classical biological applications of discrete-time Markov chain models and branching processes are illustrated including a random walk model, simple birth and death process, and epidemic process. (nimbios.org)
  • Abstract: Some applications of continuous-time Markov chains are illustrated including models for birth-death processes, competition, predation, and epidemics. (nimbios.org)
  • Abstract: Continuing the topic of efficient simulation techniques for stochastic processes, this presentation includes a full illustration of a study case involving a birth-death process and outline current, promising research avenues involving the interaction between stochastic processes modeling and modern statistical methods for Markov chains. (nimbios.org)
  • The book ends with a brief survey of the general theory of Markov processes. (ams.org)
  • It covers the development of basic properties and applications of Poisson processes and Markov chains in discrete and continuous time. (rit.edu)
  • Also, regarding the joint distribution, Wehrly and Johnson (1980) proposed a bivariate circular distribution which is related to a family of Markov processes on the circle. (uni.lu)
  • Markov chains are a type of discrete stochastic processes where the probability of event only depends on the last past event. (quantdare.com)
  • The name comes from the Russian mathematician A. Markov who, in 1913, introduced this concept when he was making an statistical investigation in poetry [4]. (quantdare.com)
  • This time we will be using Markov chains to describe the processes of the returns of the highest price of each day . (quantdare.com)
  • For admission to the course knowledge equivalent to the courses MASA01, Mathematical Statistics: Basic Course, 15 credits and MASC03, Markov processes, 7.5 credits are required together with English B. (lu.se)
  • You will study concepts such as the Ito stochastic integral with respect to a continuous martingale and related stochastic differential equations. (edu.au)
  • Special attention will be given to the classical notion of the Brownian motion, which is the most celebrated and widely used example of a continuous martingale. (edu.au)
  • A useful martingale is introduced for analyzing the stalionary law of the controlled process as well as the total expected discounted cost. (haifa.ac.il)
  • S.M. Stopped process 434 14.5 Classical examples of Martingale reasoning 439 14.6 Convergence theorems. (paradisegolf.net)
  • In this paper we study the q-optimal martingale measures using the Semimartingale Backward Equations (SBE for short) introduced by Chitashvili [3]. (123dok.com)
  • This effectively changes stochastic differential equation of the process by replacing standard Brownian motion with scaled Brownian motion. (vu.lt)
  • Here we develop a theoretical framework to model the hydrodynamic interactions between the tracer and the active swimmers, which shows that the tracer follows a non-Markovian coloured Poisson process that accounts for all empirical observations. (qmul.ac.uk)
  • The Fall 2019 version of the course was an experimental "Data Science flavored" course, which included a few extra topics (confidence intervals, moment generating function, Poisson process) and included an additional Python-based lab component (through Jupyter Notebooks) devoted to real-world data sets and their analysis using probabilistic ideas. (ucsd.edu)
  • However, in the case of a Poisson process, an infinite volume of sites are allocated to centers further away than ξ . (projecteuclid.org)
  • In this paper, we study multidimensional generalized BSDEs that have a monotone generator in a general filtration supporting a Brownian motion and an independent Poisson random measure. (vmsta.org)
  • This example is at odds with models built using fractional Brownian motion (fBm). (mdpi.com)
  • We derive fractional Brownian motion and stochastic processes with multifractal properties using a framework of network of Gaussian conditional probabilities. (harvard.edu)
  • This leads to the derivation of new representations of fractional Brownian motion. (harvard.edu)
  • Not only does this allows us to derive fractional Brownian motion, we can introduce extensions with multifractal flavour. (harvard.edu)
  • Scaled Brownian motion is a Markovian model mimicking long-range memory properties of fractional Brownian motion, such as anomalous diffusion and first passage time distribution. (vu.lt)
  • Here we have shown that similar power-law scaling behavior can be obtained both from a point process driven by the fractional Gaussian noise, and from a nonlinear Markovian point process. (vu.lt)
  • The linear fractional stable motion generalizes two prominent classes of stochastic processes, namely stable Levy processes, and fractional Brownian motion. (uni.lu)
  • We study a stylized model consisting of a superposition of independent linear fractional stable motions and our focus is on parameter estimation of the model. (uni.lu)
  • The conditions for consistency turn out to be sharp for two prominent special cases: (i) for Levy processes, i.e. for the estimation of the successive Blumenthal-Getoor indices, and (ii) for the mixed fractional Brownian motion introduced by Cheridito. (uni.lu)
  • 3 Lectures on Non-Gaussian fractional and multifractional processes. (uni-ulm.de)
  • Fractional processes play an important role in modeling the long-range dependence property. (uni-ulm.de)
  • The most popular among fractional processes is the Gaussian one, viz. (uni-ulm.de)
  • the fractional Brownian motion. (uni-ulm.de)
  • Lastly, it has Gaussian distribution, so its tails are extremely light.In view of these drawbacks, various generalizations of fractional processes are studied in the literature. (uni-ulm.de)
  • On this way, some multifractional counterparts of fractional Brownian motion were proposed. (uni-ulm.de)
  • The third drawback can be removed by considering fractional and multifractional processes which have heavier tails than that of the Gaussian distribution. (uni-ulm.de)
  • Fractional Brownian motion: definition and basic properties. (uni-ulm.de)
  • Fractional and multifractional stable processes. (uni-ulm.de)
  • We introduce a new Gaussian process, a generalization of both fractional and subfractional Brownian motions, which could serve as a good model for a larger class of natural phenomena. (vmsta.org)
  • The aim of this seminar is to give an introduction to stochastic differential equations (SDEs) and its application. (uni-ulm.de)
  • Stochastic Differential Equations (SDEs) have become a standard tool to model differential equation systems subject to noise. (lu.se)
  • Treating practical problems requires analytic techniques to understand and investigate properties of SDEs and stochastic numerical methods to compute quantities of interest, where the latter and the former often go hand in hand. (lu.se)
  • Abstract: A procedure is described for deriving a stochastic differential equation (SDE) from an associated discrete stochastic model. (nimbios.org)
  • Stochastic differential equation systems are derived for several population problems. (nimbios.org)
  • We illustrate the usefulness of our analysis by explicitly computing dynamic mean-variance portfolios under various stochastic investment opportunities in a straightforward way, which does not involve solving a Hamilton-Jacobi-Bellman differential equation. (kipdf.com)
  • Our asymptotic theory is based on new limit theorems for multiscale moving average processes. (uni.lu)
  • Non-central limit theorems and Rosenblatt process. (uni-ulm.de)
  • For the first example consider the standard Wiener process. (maplesoft.com)
  • The previous command created a new Maple variable representing the standard Wiener process. (maplesoft.com)
  • In 1923 'mathematical' Brownian motion was introduced by the Mathematician Norbert Wiener, who showed how to construct a random function B(t) with those properties. (warwick.ac.uk)
  • This mathematical object (also called the Wiener process) is the subject of this module. (warwick.ac.uk)
  • Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. (aimsciences.org)
  • Lévy-processes) or jump-diffusions? (stackexchange.com)
  • This includes processes for modeling equity prices, mean-reverting processes, pure jump processes, jump diffusions as well as multi-variate Ito processes. (maplesoft.com)
  • We consider the problem of joint parameter estimation for drift and volatility coefficients of a stochastic McKean-Vlasov equation and for the associated system of interacting particles. (uni.lu)
  • The course prepares students to engage in activities necessary for independent mathematical research and introduces students to a broad range of active interdisciplinary programs related to applied mathematics. (rit.edu)
  • The main result of this paper consists of constructing each increment of the process from two-dimensional gaussian noise inside the light-cone of each seperate increment. (harvard.edu)
  • The diffusion process followed by a passive tracer in prototypical active media, such as suspensions of active colloids or swimming microorganisms, differs considerably from Brownian motion, as revealed by a greatly enhanced diffusion coefficient and non-Gaussian statistics of the tracer displacements. (qmul.ac.uk)
  • The phenomenon has later been related in Physics to the diffusion equation, which led Albert Einstein in 1905 to postulate certain properties for the motion of an idealized 'Brownian particle' with vanishing mass: - the path t↦B(t) of the particle should be continuous, - the displacements B(t+Δt)−B(t) should be independent of the past motion, and have a Gaussian distribution with mean 0 and variance proportional to Δt. (warwick.ac.uk)
  • These processes are also Gaussian, so they do not solve the light tails problem. (uni-ulm.de)
  • In this course two kinds of processes will be considered: stable processes (having heavy tails) and square Gaussian processes (having intermediate tails). (uni-ulm.de)
  • This workshop is a continuation of a series in Applied Probability held at Carleton University with the intention to cover topics as suggested by the title of the workshop as well as important themes in diverse areas of applied probability, such as asymptotics, performance, rare event simulation, stochastic modelling, queueing theory, internet traffic, wireless network resource allocation, and optimization algorithms. (utoronto.ca)
  • By completing this unit, you will learn how to rigorously describe and tackle the evolution of random phenomena using continuous time stochastic processes. (edu.au)
  • Stochastic processes are mathematical models for random phenomena evolving in time or space. (williams.edu)
  • While the limiting process depends only on the integrated variance of the driving field, the diverging constants appearing in the definition of the reference frame also depend on higher order moments. (projecteuclid.org)
  • mar-tingale measure is a generalization of the variance-optimal marmar-tingale measure introduced by Schweizer [37], which corresponds to the case q = 2. (123dok.com)
  • The relationship between discrete-time and continuous-time processes are illustrated in these examples. (nimbios.org)
  • This theory provides a powerful unifying structure that brings together both the theory of discrete random variables and the theory of continuous random variables that were introduced earlier in your studies. (edu.au)
  • Abstract: The definition and some basic properties of Brownian motion are introduced. (nimbios.org)
  • Abstract: Brownian motion is widely used as a model of diffusion in equilibrium media throughout the physical, chemical and biological sciences. (qmul.ac.uk)
  • You will also gain a deep knowledge about stochastic integration, which is an indispensable tool to study problems arising, for example, in Financial Mathematics. (edu.au)
  • A cylindrical Lévy process does not enjoy a cylindrical version of the semimartingale decomposition which results in the need to develop a completely novel approach to stochastic integration. (projecteuclid.org)
  • You will be introduced to the fundamental concept of a measure as a generalisation of the notion of length and Lebesgue integration which is a generalisation of the Riemann integral. (edu.au)
  • MATH 286: Stochastic Integration & Stochastic Differential Equations. (ucsd.edu)
  • In Fall 2022 and 2018, I taught Math 286, an advanced graduate course on stochastic integration and stochastic differential equations. (ucsd.edu)
  • We loosely followed the textbook "Introduction to Stochastic Integration" by Chung and Williams. (ucsd.edu)
  • In addition, students will be introduced to particular mathematical models of various types: stochastic models, PDE models, dynamical system models, graph-theoretic models, algebraic models, and perhaps other types of models. (rit.edu)
  • Commonly used numerical procedures are described for computationally solving systems of stochastic differential equations. (nimbios.org)
  • In the mid to late 20th-century, ideas of algorithmic information theory introduced new dimensions to the field via the concept of algorithmic randomness. (wikipedia.org)
  • Finally, the basic theory of stochastic differential equations are introduced. (nimbios.org)
  • I can say that this book is a set of very well-written lecture notes, and it is organized as a clear synthesis of the theory of continuous-time stochastic processes with many examples and with plenty of exercises. (ams.org)
  • Students will also learn how mathematical theory, closed-form solutions for special cases, and computational methods should be integrated into the modeling process in order to provide insight into application fields and solutions to particular problems. (rit.edu)
  • This course introduces the fundamental concepts of graph theory. (rit.edu)
  • It covers a large spectrum ranging from probability theory to stochastic financial models. (stackexchange.com)
  • The theory predicts a long-lived Lévy flight regime of the loopy tracer motion with a non-monotonic crossover between two different power-law exponents. (qmul.ac.uk)
  • This unit introduces the students to modern probability theory (based on measure theory) that was developed by Andrey Kolmogorov. (edu.au)
  • Probability Theory and Stochastic Process Textbook PDF Free Download. (paradisegolf.net)
  • This 'physical' Brownian motion can be understood via the kinetic theory of heat as a result of collisions with molecules due to thermal motion. (warwick.ac.uk)
  • Over the last century, Brownian motion has turned out to be a very versatile tool for theory and applications with interesting connections to various areas of mathematics, including harmonic analysis, solutions to PDEs and fractals. (warwick.ac.uk)
  • Even though it is almost 100 years old, Brownian motion lies at the heart of deep links between probability theory and analysis, leading to new discoveries still today. (warwick.ac.uk)
  • D. R. Cox and H. D. Miller, The Theory of Stochastic Processes , CRC Press, 1977. (aimsciences.org)
  • Igor Kortchemski, A predator-prey SIR type dynamics on large complete graphs with three phase transitions, Stochastic Processes and their Applications, 10.1016/j.spa.2014.10.005, 125, 3, (886-917), (2015). (paradisegolf.net)
  • 2. UNIT VII 3 Deterministic versus probabilistic models A deterministic model can be used for a physical quantity and the process generating it provided sufficient information is available about the initial state and the dynamics of the process generating the physical quantity. (paradisegolf.net)
  • ECE-GY 6303: Probability and Stochastic Processes Course Outline by lecture (September 4, 2019 - December 20, 2019) Prof. Unnikrishna Pillai Electrical and Computer Engineering Tandon School of Engineering, NYU 370 Jay St, Room #8.03 [email protected] Lecture Room/Time: 370 Jay St/Room 202/Wed 3.20-5.50PM 1. (paradisegolf.net)
  • From a theoretical point of view (and under certain assumptions) one could model this situation by saying that the movement of the man is the trajectory of a (2 dimensional) Brownian motion. (uni-ulm.de)
  • Furthermore we will show that this probability is strictly less than one if one considers the same scenario with a ball in three dimensions and a 3 dimensional Brownian motion. (uni-ulm.de)
  • One can imagine the 3 dimensional brownian motion (under certain assumptions) as a randomly moving bird. (uni-ulm.de)
  • In this paper, two constraints for appropriate determination of the location of collocation and source points in the MFS for two-dimensional problems are introduced. (global-sci.com)
  • We aim to introduce useful thoughts and feasible procedures for finding asymptotic behaviors of the rare events in such multidimensional (mainly two dimensional) processes. (utoronto.ca)
  • In this work, we introduce a stochastic integral for random integrands with respect to cylindrical Lévy processes in Hilbert spaces. (projecteuclid.org)
  • The integral process is characterised as an adapted, Hilbert space valued semimartingale with càdlàg trajectories. (projecteuclid.org)
  • The picture shows the integral of the Browninan Motion integrated with respect to itself. (uni-ulm.de)
  • It also includes fully worked examples and as such serves as a tutorial on MRI analysis with R, from which the readers can derive their own data processing scripts. (wias-berlin.de)
  • AbstractStarting from the symmetric reduction of Cauchy bi-orthogonal polynomials, we derive the Toda equation of CKP type (or the C-Toda lattice) as well as its Lax pair by introducing the time flow. (tsinghua.edu.cn)
  • The objectives of the text are to introduce students to the standard concepts and methods of stochastic modeling, to illustrate the rich diversity of applications of stochastic processes in the applied sciences, and to provide exercises in the application of simple stochastic analysis to realistic problems. (freecomputerbooks.com)
  • This course introduces the basic concepts and techniques of stochastic processes used to construct models for a variety of problems of practical interest. (williams.edu)
  • Its aim is to introduce the basic concepts and problems of securities markets and to develop theoretical frameworks and computational tools for pricing financial products and hedging the risk associated with them. (edu.au)
  • Basic concepts in machine learning till also be introduced. (lu.se)
  • STOCHASTIC PROCESSES - SPECTRAL CHARACTERISTICS : The Power Spectrum: Properties, Relationship between Power Spectrum and Autocorrelation Function, The Cross-Power Density Spectrum, Properties, Relationship between Cross-Power Spectrum and Cross-Correlation Function. (paradisegolf.net)
  • Firstly, its increments are stationary, which does not allow to model processes whose properties vary essentially as time flows. (uni-ulm.de)
  • Secondly, it is self-similar, so has similar properties on different time scales, yet only few real-world processes have this property. (uni-ulm.de)
  • Distributional and pathwise properties of multifractional stable processes. (uni-ulm.de)
  • We study its main stochastic properties and some increments characteristics. (vmsta.org)
  • STOCHASTIC PROCESSES - TEMPORAL CHARACTERISTICS : The Stochastic Process Concept, Classification of Processes, Deterministic and Nondeterministic Processes, Distribution and Density Functions, concept of Stationarity and Statistical Independence. (paradisegolf.net)
  • We prove the strong convergence of the spectrum of kinetic Brownian motion to the spectrum of base Laplacian for all compact Riemannian manifolds. (tsinghua.edu.cn)
  • Here we introduce scaled voter model, a generalization of voter model in which herding intensity parameter is a power-law function of time. (vu.lt)
  • Stochastic processes play a key role in modelling the behavior over time of many financial assets. (quantdare.com)
  • The process of adaptation occurs via modifications of the genotype or behavior of an individual or group. (complexityexplorer.org)
  • Some statistical methods were introduced, useful for data fitting and testing of stochastic models. (nimbios.org)
  • Stable random variables and processes. (uni-ulm.de)
  • We prove that if Ξ is obtained from a translation-invariant point process, then there is a unique fair allocation which is stable in the sense of the Gale-Shapley marriage problem. (projecteuclid.org)
  • These form a natural extension of the traditional M/G/1 queues, but allow for a substantial additional generality, as they also include refelected Brownian motion and queues with alpha-stable input. (utoronto.ca)
  • We are interested in evaluating the stationary probabilities of rare events which occur in stochastic networks, provided they are stable. (utoronto.ca)
  • Further topics such as renewal processes, reliability and Brownian motion may be discussed as time allows. (rit.edu)
  • The book starts with a short introduction to MRI and then examines the process of reading and writing common neuroimaging data formats to and from the R session. (wias-berlin.de)
  • This course is an introduction to stochastic processes and their various applications. (rit.edu)
  • Welcome to the Web site for Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers, 3rd Edition by Roy D. Yates and David J. Goodman. (paradisegolf.net)
  • Probability and Stochastic Processes A Friendly Introduction for Electrical and Computer Engineers Third Edition International Students' Version QUIZ SOLUTIONS Roy D. Yates, David J. Goodman, David Famolari April 30, 2014 1. (paradisegolf.net)
  • Nous proposons une classification des données manquantes en deux catégories Missing At Random et Not Missing At Random pour les modèles à variables latentes suivant le modèle décrit par D. Rubin. (inrae.fr)
  • We believe our method has widely application in the collective motions and active particle systems. (tsinghua.edu.cn)
  • Then, the parameter estimation process is illustrated using computer intensive methods, such as MCMC and other, recent developments requiring efficient simulation techniques for such stochastic processes. (nimbios.org)
  • Furthermore we introduce a class of joint circular distributions from the higher order spectra of time series, which can describe very general joint circular distributions. (uni.lu)
  • Because we introduced very general joint circular distributions, we can discuss the problem of copula for them. (uni.lu)
  • This tutorial was designed to introduce selected topics in stochastic models with an emphasis on biological applications. (nimbios.org)
  • Models arising in physics and engineering are introduced. (rit.edu)
  • The class of critical bootstrap percolation models in two dimensions was recently introduced by Bollobás, Smith and Uzzell, and the critical threshold for percolation was determined up to a constant factor for all such models by the authors of this paper. (projecteuclid.org)
  • Typically models of 1/f noise in solid state matter involve point processes, or telegraph-like processes. (vu.lt)
  • The aim of this course is to introduce students to common deep learnings architectues such as multi-layer perceptrons, convolutional neural networks and recurrent models such as the LSTM. (lu.se)
  • The models we focus on are formulated as stochastic differential equations (SDE:s). (lu.se)
  • There is a wide spectrum of problems in these fields, which are described using random processes that evolve with time. (edu.au)
  • Usually, the random variables that make up the process are not independent from each other. (quantdare.com)
  • In this study, we introduce a diffusion approximation to this system. (haifa.ac.il)
  • The first passage time density of a diffusion process to a time varying threshold is of primary interest in different fields. (aimsciences.org)