VO Stochastic Processes - Zusammenfassung Buch

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General

(Z(t, w))_t \in \mathbb{R}^+ \, is a collection of random variables defined on some probability space (\Omega, \mathcal A, \mathbb R)\,

t fixed Z(t, w)\, is a random variable
w fixed  t \mapsto Z(t, w)\, is a function on \mathbb{R}^+\, (trajectory or path of the stoch. process)
t discrete discrete time porcesses, most import ones are the Markov processes

Ergodicity

Ergodicity allows us to extract all the information we need from 1 trajectory because

\hat{H}_T = \frac 1 T \int_0^T H(Z(t, w)) dt \rightarrow \operatorname{E}(H(Z(.))) \,

because something like LLN is valid.

The lower the ergodicity coefficient the faster is convergence

Markov Chains

  • The process is called a homogenous Markov Chain if there exists
P = (p_{ij})\,

such that

P(M(n) = z_j | M(n-1) = z_i) = p_{ij} \qquad \forall n
  • The probablility KLAUS of a homogenous Markov Chain is determined by the distribution of M(0)\, (starting distribution) and the transition matrix P\,.
  • If the process is startetd with a starting distribution \gamma = (\gamma_1, \gamma_2, \ldots, \gamma_3)\, and P(M(0) = z_i) = \gamma_i\,

then the distribution of M(1)\, is

\begin{align}
P(M(1) = z_j) 
& = \sum{i=1}^s P(M(1) = z_j| M(0) = z_i)  P(M(0) = z_i) \\
& = \sum{i=1}^s \gamma_i p_{ij}
\end{align}
  • M(n) = \gamma P^n \,
  • Expected number of visists in state z_j\, if started in z_i\, is:
\nu_{ij} = \sum_{n=0}^\infty p_{ij}^n\,
  • Putting values together in matrix V= (v_{ij} = \sum_{n=0}^\infty P^n) \,

Definitions

  • z_i\, is reachable from z_j\, if there exists n > 0\, with p_{ij}^n > 0\,: z_i \rightsquigarrow z_j\,
  • Commuting states: z_i \leftrightsquigarrow z_j\,
  • Let G_\alpha, G_\beta, \ldots\, be the equivalnce classes of commuting states. We introduce a partial ordering for these classes by saying that
G_\alpha\, preceedes G_\beta\, (in symbol G_\alpha \succ G_\beta\,) if for all z_i \in G_\alpha\, and all z_j \in G_\beta\, z_i \rightsquigarrow z_j\,

(there may also exist incomparable classes)

  • A class is called transient, if it preceeds another class. Otherwise the class is called a maximal class.
  • A state is called recurrent if the chain which is started in z_i\, returns with probability 1 to z_i\,.


KLAUS