Continuous Markov

Definition

• \((\Omega, \mathcal{F}, P, (\mathcal{F}_{t})_{t \ge 0})\),
  • probability space with fltration

Def(Markov Process)

• $\mu$
  • Borel prob. measure over $\mathbb{R}^{d}$

$(X_{t})_{t \ge 0}$ is said to be markov process with initial distribution $\mu$ over \((\Omega, \mathcal{F}, P)\) if

1. \(P(X_{0} \in \Gamma) = \mu(\Gamma) \quad (\forall \Gamma \in \mathcal{B}(\mathbb{R}^{d})\),
2. If \(s, t \ge 0\), \(\Gamma \in \mathcal{B}(\mathbb{R}^{d})\),

\[P(X_{s+t} \in \Gamma \mid \mathcal{F}_{s}) = P(X_{s+t} \in \Gamma \mid X_{s}) \quad \mathrm{a.s.}\]

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Def(Markov family)

• $x \in \mathbb{R}^{d}$,
  • given

\((X_{t}^{x})_{t \ge 0}\) is said to be time-homogeneous markov family if

• (1) the following function $p$ is borel measurable as a function of $x$

\[p(t, x, \Gamma) := P(X_{t}^{x} \in \Gamma) \quad ( t \ge 0, x \in \mathbb{R}^{d}, \Gamma \in \mathcal{B}(\mathbb{R}^{d}) ) .\]

• (2) Satisfy

\[P(X_{0}^{x} = x) = 1 .\]

• (3) For $s, t \ge 0 $, $x \in \mathbb{R}^{d}$, $\Gamma \in \mathcal{B}(\mathbb{R}^{d})$,

\[P(X_{s+t}^{x} \in \Gamma \mid \mathcal{F}_{s}) = p(t, X_{s}^{x}, \Gamma) \quad \mathrm{a.s.}\]

The function $p$ defined in (1) is called transition function.

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Reference

• Markov chain - Wikipedia