scholarly journals A probability space of continuous-time discrete value stochastic process with Markov property

2016 ◽  
Vol 12 (3) ◽  
pp. 5975-5991
Author(s):  
Miloslawa Sokol
Keyword(s):  
Stochastic Process ◽  
Probability Theory ◽  
Stochastic Processes ◽  
Continuous Time ◽  
Probability Space ◽  
Measure Theory ◽  
Markov Property ◽  
Finite Measure ◽  
Physical Models

Getting acquainted with the theory of stochastic processes we can read the following statement: "In the ordinary axiomatization of probability theory by means of measure theory, the problem is to construct a sigma-algebra of measurable subsets of the space of all functions, and then put a finite measure on it". The classical results for limited stochastic and intensity matrices goes back to Kolmogorov at least late 40-s. But for some infinity matrices the sum of probabilities of all trajectories is less than 1. Some years ago I constructed physical models of simulation of any stochastic processes having a stochastic or an intensity matrices and I programmed it. But for computers I had to do some limitations - set of states at present time had to be limited, at next time - not necessarily. If during simulation a realisation accepted a state out of the set of limited states - the simulation was interrupted. I saw that I used non-quadratic, half-infinity stochastic and intensity matrices and that the set of trajectories was bigger than for quadratic ones. My programs worked good also for stochastic processes described in literature as without probability space. I asked myself: did the probability space for these experiments not exist or were only set of events incompleted? This paper shows that the second hipothesis is true.

scholarly journals On the weak convergence of stochastic processes with embedded point processes

1988 ◽  
Vol 20 (2) ◽  
pp. 473-475 ◽  
Author(s):  
Panagiotis Konstantopoulos ◽  
Jean Walrand
Keyword(s):  
Stochastic Process ◽  
Weak Convergence ◽  
Stochastic Processes ◽  
Point Process ◽  
Real Line ◽  
Continuous Time ◽  
Point Processes ◽  
Mixing Condition ◽  
The Real ◽  
Palm Distribution

We consider a stochastic process in continuous time and two point processes on the real line, all jointly stationary. We show that under a certain mixing condition the values of the process at the points of the second point process converge weakly under the Palm distribution with respect to the first point process, and we identify the limit. This result is a supplement to two other known results which are mentioned below.

scholarly journals The relationship between intensity and stochastic matrices for continuous-time discrete value stochastic non-homogeneous processes with Markov property

2017 ◽  
Vol 13 (3) ◽  
pp. 7244-7256
Author(s):  
Mi los lawa Sokol
Keyword(s):  
Markov Process ◽  
Stochastic Processes ◽  
Markov Processes ◽  
Continuous Time ◽  
Markov Property ◽  
Time Dependent ◽  
Stochastic Matrices ◽  
Homogeneous Markov Process ◽  
The Relationship ◽  
Time Form

The matrices of non-homogeneous Markov processes consist of time-dependent functions whose values at time form typical intensity matrices. For solvingsome problems they must be changed into stochastic matrices. A stochas-tic matrix for non-homogeneous Markov process consists of time-dependent functions, whose values are probabilities and it depend on assumed time pe- riod. In this paper formulas for these functions are derived. Although the formula is not simple, it allows proving some theorems for Markov stochastic processes, well known for homogeneous processes, but for non-homogeneous ones the proofs of them turned out shorter.

Stochastic Processes in Continuous Time

2021 ◽  
pp. 611-637
Author(s):  
James Davidson
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Continuous Time ◽  
Continuous Process ◽  
Markov Property ◽  
Reflection Principle ◽  
Finite Variance ◽  
The Central Limit Theorem ◽  
The Family ◽  
Strong Markov

This chapter reviews the theory of continuous-time stochastic processes, covering the concepts of adaptation, Lévy processes, diffusions, martingales, and Markov processes. Brownian motion is studied as the most important case, with properties that include the reflection principle and the strong Markov property. The technique of Skorokhod embedding is introduced, providing novel proofs of the central limit theorem and the law of the iterated logarithm. The family of processes derived from Brownian motion is reviewed and in the final section it is shown that a continuous process having finite variance and independent increments is Brownian motion.

scholarly journals Contextuality Scenarios Arising from Networks of Stochastic Processes

2016 ◽  
Vol 23 (03) ◽  
pp. 1650012
Author(s):  
Rodrigo Iglesias ◽  
Fernando Tohmé ◽  
Marcelo Auday
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Empirical Model ◽  
Joint Distribution ◽  
Probability Space ◽  
Random Variables ◽  
Empirical Models ◽  
Classical Source ◽  
Input And Output ◽  
Long Run

An empirical model is a generalization of a probability space. It consists of a simplicial complex of subsets of a class 𝒳 of random variables such that each simplex has an associated probability distribution. The ensuing marginalizations are coherent, in the sense that the distribution on a face of a simplex coincides with the marginal of the distribution over the entire simplex. An empirical model is called contextual if its distributions cannot be obtained by marginalizing a joint distribution over 𝒳. Contextual empirical models arise naturally in quantum theory, giving rise to some of its counter -intuitive statistical consequences. In this paper, we present a different and classical source of contextual empirical models: the interaction among many stochastic processes. We attach an empirical model to the ensuing network in which each node represents an open stochastic process with input and output random variables. The statistical behaviour of the network in the long run makes the empirical model generically contextual and even strongly contextual.

scholarly journals Doob: A Half-Century on

2005 ◽  
Vol 42 (01) ◽  
pp. 257-266
Author(s):  
N. H. Bingham
Keyword(s):  
Stochastic Process ◽  
Probability Theory ◽  
Seventeenth Century ◽  
Stochastic Processes ◽  
Fiftieth Anniversary ◽  
Process Theory ◽  
Dynamic Aspect ◽  
Half Century ◽  
Classic Work

Probability theory, and its dynamic aspect stochastic process theory, is both a venerable subject, in that its roots go back to the mid-seventeenth century, and a young one, in that its modern formulation happened comparatively recently - well within living memory. The year 2003 marked the seventieth anniversary of Kolmogorov's Grundbegriffe der Wahrscheinlichkeitsrechnung, usually regarded as having inaugurated modern (measure-theoretic) probability theory. It also marked the fiftieth anniversary of Doob's Stochastic Processes. The profound and continuing influence of this classic work prompts the present piece.

scholarly journals On the weak convergence of stochastic processes with embedded point processes

1988 ◽  
Vol 20 (02) ◽  
pp. 473-475
Author(s):  
Panagiotis Konstantopoulos ◽  
Jean Walrand
Keyword(s):  
Stochastic Process ◽  
Weak Convergence ◽  
Stochastic Processes ◽  
Point Process ◽  
Real Line ◽  
Continuous Time ◽  
Point Processes ◽  
Mixing Condition ◽  
The Real ◽  
Palm Distribution

We consider a stochastic process in continuous time and two point processes on the real line, all jointly stationary. We show that under a certain mixing condition the values of the process at the points of the second point process converge weakly under the Palm distribution with respect to the first point process, and we identify the limit. This result is a supplement to two other known results which are mentioned below.

scholarly journals A structure theorem for stochastic processes indexed by the discrete hypercube

2021 ◽  
Vol 9 ◽  
Author(s):  
Pandelis Dodos ◽  
Konstantinos Tyros
Keyword(s):  
Stochastic Process ◽  
Positive Integer ◽  
Stochastic Processes ◽  
Probability Space ◽  
Structure Theorem ◽  
Structural Information ◽  
Cartesian Product ◽  
Stationarity Condition ◽  
Finite Set ◽  
As If

Abstract Let A be a finite set with , let n be a positive integer, and let $A^n$ denote the discrete $n\text {-dimensional}$ hypercube (that is, $A^n$ is the Cartesian product of n many copies of A). Given a family $\langle D_t:t\in A^n\rangle $ of measurable events in a probability space (a stochastic process), what structural information can be obtained assuming that the events $\langle D_t:t\in A^n\rangle $ are not behaving as if they were independent? We obtain an answer to this problem (in a strong quantitative sense) subject to a mild ‘stationarity’ condition. Our result has a number of combinatorial consequences, including a new (and the most informative so far) proof of the density Hales-Jewett theorem.

scholarly journals Doob: A Half-Century on

2005 ◽  
Vol 42 (1) ◽  
pp. 257-266 ◽  
Author(s):  
N. H. Bingham
Keyword(s):  
Stochastic Process ◽  
Probability Theory ◽  
Seventeenth Century ◽  
Stochastic Processes ◽  
Fiftieth Anniversary ◽  
Process Theory ◽  
Dynamic Aspect ◽  
Half Century ◽  
Classic Work

Probability theory, and its dynamic aspect stochastic process theory, is both a venerable subject, in that its roots go back to the mid-seventeenth century, and a young one, in that its modern formulation happened comparatively recently - well within living memory. The year 2003 marked the seventieth anniversary of Kolmogorov'sGrundbegriffe der Wahrscheinlichkeitsrechnung, usually regarded as having inaugurated modern (measure-theoretic) probability theory. It also marked the fiftieth anniversary of Doob'sStochastic Processes. The profound and continuing influence of this classic work prompts the present piece.

On Two Hypotheses in Economic Analysis of Stochastic Processes

2018 ◽  
Vol 8 (8) ◽  
pp. 2391
Author(s):  
Dmitry Aleksandrovich ENDOVITSKY ◽  
Valery Vladimirovich DAVNIS ◽  
Viacheslav Vladimirovich KOROTKIKH
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Probability Space ◽  
Econometric Modeling ◽  
Model Errors ◽  
Expectations Hypothesis ◽  
Additional Information ◽  
Model Residuals ◽  
Econometric Approach ◽  
Probabilistic Nature

Purpose: Development of the apparatus of the stochastic processes econometric modeling. Discussion: The authors identify risk component in the dynamics of stochastic processesin the economy. Theoretical justification of the alternative and proportional expectations is usedto make probabilistic nature of the risk. Results: The authors suggest stochastic process decomposition based on econometric approach to allocate a probability space of risks, and to identify shocks realizations that lie beyond the boundary of this space. Proportional expectations hypothesis distinguished two types of the event influence on the stochastic process realization: continuous (risk) and discrete (shock). The authors suggest model errors and residualsas the main source of information for the identification of the probability space of risks. The technique of econometric modeling of the price and return processes on stock market under theconditions of the proposed hypotheses is considered in the empirical part of the study. F-testresults have not disproved the statement that the model residuals provide additional information about the simulated rate in the case of lack of relevant factors.

A computation of the Littlewood exponent of stochastic processes

1988 ◽  
Vol 103 (2) ◽  
pp. 367-370 ◽  
Author(s):  
Ron C. Blei ◽  
J.-P. Kahane
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Total Variation ◽  
Probability Space ◽  
Complex Measure ◽  
Bona Fide ◽  
Image Position

A stochastic process X = {X(t): t ∈ [0, 1]} on a probability space (Ω, , ℙ) is said to have finite expectation if the function defined on the measureable rectangles in Ω × [0, 1] byfor A ∈ and (s, t) ⊂ [0, 1] gives rise to a complex measure in each of its two coordinates (see [1], definition 1·1). Equivalently, X has finite expectation ifis finite. The function defined by (1), effectively a generalization of the Doléans measure (see e.g. [4] pp. 33–35), is extendible to a bona fide complex measure on Ω × [0, 1] if and only if its ‘total variation’

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