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 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.

scholarly journals Limiting stochastic processes of shift-periodic dynamical systems

2019 ◽  
Vol 6 (11) ◽  
pp. 191423
Author(s):  
Julia Stadlmann ◽  
Radek Erban
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Discrete Time ◽  
Real Line ◽  
Continuous Time ◽  
Initial Distribution ◽  
Dynamical Behaviour ◽  
One Dimensional

A shift-periodic map is a one-dimensional map from the real line to itself which is periodic up to a linear translation and allowed to have singularities. It is shown that iterative sequences x n +1 = F ( x n ) generated by such maps display rich dynamical behaviour. The integer parts ⌊ x n ⌋ give a discrete-time random walk for a suitable initial distribution of x 0 and converge in certain limits to Brownian motion or more general Lévy processes. Furthermore, for certain shift-periodic maps with small holes on [0,1], convergence of trajectories to a continuous-time random walk is shown in a limit.

The Brownian bridge as a flat integral

1989 ◽  
Vol 106 (2) ◽  
pp. 343-354 ◽  
Author(s):  
Nigel Cutland
Keyword(s):  
Brownian Motion ◽  
Brownian Bridge ◽  
Continuous Process ◽  
Wiener Measure ◽  
The Family

The family of Brownian bridge processes (ba)a∈R has a number of characterizations, the most fundamental being that ba: [0,1] → ℝ is Brownian motion conditioned to be at the point a at time 1. Equivalently, ba is a continuous process whose law Wa is that of Wiener measure conditioned on the set of paths with x1 = a. These ideas are not so easy to make precise, so that more down to earth and workable characterizations of the Brownian bridge such as the following are often used in practice (see [6] for example)

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.

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