On the Markov Property

Abstract

Let $\left( {\Omega, \,\mathcal{A},\,\mathbb{P}} \right)$ be a probability space with some filtration $\left( {\mathcal{F}_t } \right)_{t \geqslant 0}$ and a d-dimensional adapted stochastic process $X = \left( {X_t } \right)_{t \geqslant 0}$, i.e., each X_t is $\mathcal{F}_t$ measurable.We write $\mathcal{B}\left( {\mathbb{R}^d } \right)$ for the Borel sets and set $F_\infty : = \sigma \left( { \cup _{t \geqslant 0} F_t} \right)$.