Stochastic Processes in Discrete Time

1998 ◽  
pp. 67-82
Author(s):  
Nicholas H. Bingham ◽  
Rüdiger Kiesel
Keyword(s):  
Stochastic Processes ◽  
Discrete Time

scholarly journals Markov Stochastic Processes in Biology and Mathematics -- the Same, and yet Different

2018 ◽  
Vol 14 (1) ◽  
pp. 7540-7559
Author(s):  
MI lOS lAWA SOKO
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Discrete Time ◽  
Stochastic Matrix ◽  
Random Number Generator ◽  
Point Of View ◽  
State Transitions ◽  
Similar Principle ◽  
Probabilistic Space ◽  
And Mathematics

Virtually every biological model utilising a random number generator is a Markov stochastic process. Numerical simulations of such processes are performed using stochastic or intensity matrices or kernels. Biologists, however, define stochastic processes in a slightly different way to how mathematicians typically do. A discrete-time discrete-value stochastic process may be defined by a function p : X0 × X → {f : Î¥ → [0, 1]}, where X is a set of states, X0 is a bounded subset of X, Î¥ is a subset of integers (here associated with discrete time), where the function p satisfies 0 < p(x, y)(t) < 1 and  EY p(x, y)(t) = 1. This definition generalizes a stochastic matrix. Although X0 is bounded, X may include every possible state and is often infinite. By interrupting the process whenever the state transitions into the X −X0 set, Markov stochastic processes defined this way may have non-quadratic stochastic matrices. Similar principle applies to intensity matrices, stochastic and intensity kernels resulting from considering many biological models as Markov stochastic processes. Class of such processes has important properties when considered from a point of view of theoretical mathematics. In particular, every process from this class may be simulated (hence they all exist in a physical sense) and has a well-defined probabilistic space associated with it.

Stochastic Processes in Discrete Time

2010 ◽  
pp. 55-65
Author(s):  
Jürgen Franke ◽  
Wolfgang Karl Härdle ◽  
Christian Matthias Hafner
Keyword(s):  
Stochastic Processes ◽  
Discrete Time

Stochastic Processes in Discrete Time

2004 ◽  
pp. 43-52
Author(s):  
Jürgen Franke ◽  
Wolfgang Härdle ◽  
Christian M. Hafner
Keyword(s):  
Stochastic Processes ◽  
Discrete Time

Criteria for the non-ergodicity of stochastic processes: application to the exponential back-off protocol

1987 ◽  
Vol 24 (02) ◽  
pp. 347-354 ◽  
Author(s):  
Guy Fayolle ◽  
Rudolph Iasnogorodski
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
Stochastic Processes ◽  
State Space ◽  
Discrete Time ◽  
Random Variables ◽  
Countable State Space ◽  
Countable State ◽  
New Criteria

In this paper, we present some simple new criteria for the non-ergodicity of a stochastic process (Yn ), n ≧ 0 in discrete time, when either the upward or downward jumps are majorized by i.i.d. random variables. This situation is encountered in many practical situations, where the (Yn ) are functionals of some Markov chain with countable state space. An application to the exponential back-off protocol is described.

Controller Design for Discrete-Time Stochastic Processes With Nonquadratic Loss

Technometrics ◽  
2006 ◽  
Vol 48 (3) ◽  
pp. 361-372
Author(s):  
Michael G Forbes ◽  
J. Fraser Forbes ◽  
Martin Guay
Keyword(s):  
Stochastic Processes ◽  
Discrete Time ◽  
Controller Design
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