Dependent jump processes with coupled Lévy measures

I present a simple method for the modeling and simulation of dependent positive jump processes through a series representation. Each constituent process is represented by a series whose terms are equal to a transformation of the jump times of a standard Poisson process. The transformations are given by the inverses of the respective marginal Lévy tail mass integral functions. The dependence between the various constituent processes is given by a probabilistic copula for the inter-arrival times of the various standard Poisson processes.Lévy copulas, Copulas, Lévy processes, Monte-Carlo simulations

Probabilistic properties of detrended fluctuation analysis for Gaussian processes

Detrended fluctuation analysis (DFA) is one of the most widely used tools for the detection of long-range dependence in time series. Although DFA has found many interesting applications and has been shown to be one of the best performing detrending methods, its probabilistic foundations are still unclear. In this paper, we study probabilistic properties of DFA for Gaussian processes. Our main attention is paid to the distribution of the squared error sum of the detrended process. We use a probabilistic approach to derive general formulas for the expected value and the variance of the squared fluctuation function of DFA for Gaussian processes. We also get analytical results for the expected value of the squared fluctuation function for particular examples of Gaussian processes, such as Gaussian white noise, fractional Gaussian noise, ordinary Brownian motion, and fractional Brownian motion. Our analytical formulas are supported by numerical simulations. The results obtained can serve as a starting point for analyzing the statistical properties of DFA-based estimators for the fluctuation function and long-memory parameter

Categories of Timed Stochastic Relations

AbstractStochastic behavior—the probabilistic evolution of a system in time—is essential to modeling the complexity of real-world systems. It enables realistic performance modeling, quality-of-service guarantees, and especially simulations for biological systems. Languages like the stochastic pi calculus have emerged as effective tools to describe and reason about systems exhibiting stochastic behavior. These languages essentially denote continuous-time stochastic processes, obtained through an operational semantics in a probabilistic transition system. In this paper we seek a more descriptive foundation for the semantics of stochastic behavior using categories and monads. We model a first-order imperative language with stochastic delay by identifying probabilistic choice and delay as separate effects, modeling each with a monad, and combining the monads to build a model for the stochastic language

Метод пошаговой реструктуризации имитационных моделей для исследования вероятностных технологических процессов

Рассматривается новый подход к построению имитационных моделей вероятностных технологических процессов переменной структуры. Излагается метод пошаговой реструктуризации имитационной модели вероятностного сетевого графика в режиме реального времени (инвариантное погружение соответствующей модели вероятностного технологического процесса во множество вероятностных моделей сетевого планирования с переменной структурой) для решения задачи исследования и управления вероятностными технологическими процессами. На основе метода пошаговой реструктуризации и новой версии системы автоматизации имитационного моделирования агрегатного типа предлагается способ имитации вероятностных технологических процессов, который является развитием методов решения классической проблемы синтеза оптимальных систем для вероятностных технологических процессов с изменяющейся структурой технологического цикла.Розглядається новий підхід до побудови імітаційних моделей імовірнісних технологічних процесів змінної структури. Висловлюється метод покрокової реструктуризації імітаційної моделі імовірнісного мережевого графіка в режимі реального часу (інваріантне занурення відповідної моделі імовірнісного технологічного процесу в безліч імовірнісних моделей мережевого планування зі змінною структурою) для вирішення завдання дослідження і управління імовірнісними технологічними процесами. На основі методу покрокової реструктуризації і нової версії системи автоматизації імітаційного моделювання агрегатного типу пропонується спосіб імітації імовірнісних технологічних процесів, який є розвитком методів вирішення класичної проблеми синтезу оптимальних систем для імовірнісних технологічних процесів із структурою технологічного циклу, що змінюється.New approach is examined to the construction of models of simulations of probabilistic technological processes of variable structure. The method of the incremental restructuring of simulation model of the probabilistic network graph is expounded in the real-time (invariant immersion of the proper model of probabilistic technological process in the great number of probabilistic models of the network planning with a variable structure) mode for the decision of task of research and control of probabilistic technological processes. On the basis of method of the incremental restructuring and the update version of the system of automation of imitation design of aggregate type it is offered the method of imitation of probabilistic technological processes, which is development of methods of decision of classic problem of synthesis of the optimum systems for probabilistic technological processes with changing structure of technological cycle

Topology-guided sampling of nonhomogeneous random processes

Topological measurements are increasingly being accepted as an important tool for quantifying complex structures. In many applications, these structures can be expressed as nodal domains of real-valued functions and are obtained only through experimental observation or numerical simulations. In both cases, the data on which the topological measurements are based are derived via some form of finite sampling or discretization. In this paper, we present a probabilistic approach to quantifying the number of components of generalized nodal domains of nonhomogeneous random processes on the real line via finite discretizations, that is, we consider excursion sets of a random process relative to a nonconstant deterministic threshold function. Our results furnish explicit probabilistic a priori bounds for the suitability of certain discretization sizes and also provide information for the choice of location of the sampling points in order to minimize the error probability. We illustrate our results for a variety of random processes, demonstrate how they can be used to sample the classical nodal domains of deterministic functions perturbed by additive noise and discuss their relation to the density of zeros.Comment: Published in at http://dx.doi.org/10.1214/09-AAP652 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Feynman-Kac representation of fully nonlinear PDEs and applications

The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in stochastic analysis and the emergence of the theory of backward stochastic differential equations (BSDEs), which can be viewed as nonlinear Feynman-Kac formulas. We review in this paper the main ideas and results in this area, and present implications of these probabilistic representations for the numerical resolution of nonlinear PDEs, together with some applications to stochastic control problems and model uncertainty in finance

Characterising Probabilistic Processes Logically

In this paper we work on (bi)simulation semantics of processes that exhibit both nondeterministic and probabilistic behaviour. We propose a probabilistic extension of the modal mu-calculus and show how to derive characteristic formulae for various simulation-like preorders over finite-state processes without divergence. In addition, we show that even without the fixpoint operators this probabilistic mu-calculus can be used to characterise these behavioural relations in the sense that two states are equivalent if and only if they satisfy the same set of formulae.Comment: 18 page

Smart Sampling for Lightweight Verification of Markov Decision Processes

Markov decision processes (MDP) are useful to model optimisation problems in concurrent systems. To verify MDPs with efficient Monte Carlo techniques requires that their nondeterminism be resolved by a scheduler. Recent work has introduced the elements of lightweight techniques to sample directly from scheduler space, but finding optimal schedulers by simple sampling may be inefficient. Here we describe "smart" sampling algorithms that can make substantial improvements in performance.Comment: IEEE conference style, 11 pages, 5 algorithms, 11 figures, 1 tabl

Logical Characterizations of Behavioral Relations on Transition Systems of Probability Distributions

Probabilistic nondeterministic processes are commonly modeled as probabilistic LTSs (PLTSs). A number of logical characterizations of the main behavioral relations on PLTSs have been studied. In particular, Parma and Segala [2007] and Hermanns et al. [2011] define a probabilistic Hennessy-Milner logic interpreted over probability distributions, whose corresponding logical equivalence/preorder when restricted to Dirac distributions coincide with standard bisimulation/simulation between the states of a PLTS. This result is here extended by studying the full logical equivalence/preorder between (possibly non-Dirac) distributions in terms of a notion of bisimulation/simulation defined on a LTS whose states are distributions (dLTS). We show that the well-known spectrum of behavioral relations on nonprobabilistic LTSs as well as their corresponding logical characterizations in terms of Hennessy-Milner logic scales to the probabilistic setting when considering dLTSs