The Annals of Probability

A Predictive View of Continuous Time Processes

Frank B. Knight

Full-text: Open access

Abstract

Let $X(t), 0 \leqq t$, be an $\mathscr{L} \times \mathscr{F}$-measurable process on $(\Omega, \mathscr{F}, P)$ with state space $(E, \mathscr{E})$, where $\mathscr{L}$ is the Lebesgue $\sigma$-field and $\mathscr{E}$ is countably generated. Let $\mathscr{F}(t_1, t_2), 0 \leqq t_1 < t_2 \leqq \infty$, be the $\sigma$-field generated by $\{\int^t_{t_1}f(X(s)) ds, t_1 < t < t_2, 0 \leqq f \in \mathscr{E}\}$. A new process $Z(t)$ is constructed whose values consist of conditional probabilities in the wide sense over $\mathscr{F}(t, \infty)$ given $\mathscr{F}(0, t+)$. It is shown that $Z(t)$ is a homogeneous strong-Markov process on a compact metric space, with right-continuous paths having left limits for $t > 0. Z(t)$ determines $X(t) \mathrm{wp} 1$ except for $t$ in a Lebesgue-null set. We call $Z(t)$ the prediction process of $X(t)$. Some general properties of the construction are developed, followed by two applications.

Article information

Source
Ann. Probab. Volume 3, Number 4 (1975), 573-596.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1176996302

Digital Object Identifier
doi:10.1214/aop/1176996302

Mathematical Reviews number (MathSciNet)
MR383513

Zentralblatt MATH identifier
0317.60018

JSTOR
links.jstor.org

Keywords
6040 6060 6062 Continuous time processes prediction strong Markov processes conditional probabilities in the wide sense

Citation

Knight, Frank B. A Predictive View of Continuous Time Processes. Ann. Probab. 3 (1975), no. 4, 573--596. doi:10.1214/aop/1176996302. http://projecteuclid.org/euclid.aop/1176996302.


Export citation