|
In probability theory, a Markov process is a stochastic process in which the probability distribution of the current state is conditionally independent of the path of past states, a characteristic called the Markov property.In short Markov means no memory beyond the present.
Mathematically, the Markov process is expressed as for any n and Failed to parse (Missing texvc executable; please see math/README to configure.): t_1<t_2...<t_n
,
- Failed to parse (Missing texvc executable; please see math/README to configure.): P[x(t_n) \le x_n~|~x(t)~\forall~t \le t_{n-1}] = P[x(t_n) \le x_n~|~x(t_{n-1})]\,\!
A Markov process is, as translations from Russian state, "a process without after-effect". This means that the process has no memory, save the memory of the last observed point (the 2-point transition density depends exactly on 2 points). From the standard definitions of drift and diffusion coefficients this means that those coefficients may depend at most on a single point (x, t) and on no history at all. The initial conditions cannot be remembered by the drift and diffusion coefficients of a Markov process. An Ito process, in contrast, may exhibit finite memory, the history of a finite number of past states that a trajectory has passed through. In this case the drift and diffusion coefficients contain that memory. The Chapman-Kolmogorov equation is a necessary but insufficient condition for a Markov process. A general Ito process with memory obeys a Chapman-Kolmogorov equation, as do transition densities of generalized Black-Scholes equations, whose solutions are given by a Feynman-Kac transformation on the solutions of Kolmogorov's backward time equation. Fractional Brownian motion, e.g., is not an Ito process, it has memory at the level of pair correlations. Ito processes without drift (martingales) have no memory at the level of pair correlations. This is why Ito processes are used to describe markets that are hard to beat, or are effectively efficient.
Often, the term Markov chain is used to mean a discrete-time Markov process. Also see continuous-time Markov process.
Mathematically, if X(t), t > 0, is a stochastic process, the Markov property states that
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{Pr}\big[X(t+h) = y \,|\, X(s) = x(s), \forall s \leq t\big] = \mathrm{Pr}\big[X(t+h) = y \,|\, X(t) = x(t)\big], \quad \forall h > 0.
Markov processes are typically termed (time-) homogeneous if
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{Pr}\big[X(t+h) = y \,|\, X(t) = x\big] = \mathrm{Pr}\big[X(h) = y \,|\, X(0) = x(0)\big], \quad \forall t, h > 0,
and otherwise are termed (time-) inhomogeneous (or (time-) nonhomogeneous). Homogeneous Markov processes, usually being simpler than inhomogeneous ones, form the most important class of Markov processes.
In some cases, apparently non-Markovian processes may still have Markovian representations, constructed by expanding the concept of the 'current' and 'future' states. For example, let X be a non-Markovian process. Then define a process Y, such that each state of Y represents a time-interval of states of X, i.e. mathematically,
- Failed to parse (Missing texvc executable; please see math/README to configure.): Y(t) = \big\{ X(s): s \in [a(t), b(t)] \, \big\}.
If Y has the Markov property, then it is a Markovian representation of X. In this case, X is also called a second-order Markov process. Higher-order Markov processes are defined analogously.
An example of a non-Markovian process with a Markovian representation is a moving average time series.
References
- Eric W. Weisstein, Markov process at MathWorld.
- Avner Friedman, Stochastic Differential Equations and Applications, Academic, N.Y., 1975.
- Joseph L. McCauley, Kevin E. Bassler, and Gemunu H. Gunaratne, Martingales, Detrending Data, and the Efficient Market Hypothesis, Physica A37, 202, 2008..
See also
af:Markovproses
bg:Марковски процес nl:Markovproces ja:マルコフ過程 ru:Марковский процесс uk:Марківський процес
|
|
