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 Teor. Veroyatnost. i Primenen., 1968, Volume 13, Issue 1, Pages 17–38 (Mi tvp790)

Extrapolation of multidimensional Markov processes from incomplete data

R. Sh. Liptser, A. N. Shiryaev

Moscow

Abstract: Let $(\theta_t,\eta_t)$, $t\ge0$, be a Markov process, where $\eta_t$ is the observable component and $\theta_t$ is the unobservable one. Put
$$\pi_\beta(\tau,t)=\mathbf P(\theta_\tau=\beta\mid\eta_s, s\le t),\quad\tau\ge t,$$
if $\theta_t$ takes discrete values and
$$\pi_\beta(\tau,t)=\frac{\partial\mathbf P(\theta_t\le\beta\mid\eta_s, s\le t)}{\partial\beta},\quad\tau\ge t,$$
if $\theta_\tau$ takes continuous values. When $\theta_t$, $t\ge0$, is a purely discontinuous Markov process and $\eta_t$ has the stochastic differential (5), in § 1 equations in $t$ and $\tau$ for $\pi_\beta(\tau,t)$ are deduced. In § 2 equations for the density $\pi_\beta(\tau,t)$ are obtained under the supposition that $(\theta_t,\eta_t)$ be a diffusion Markov process. In § 3 some cases of effective solving of extrapolation problems for processes regarded in § 2 are considered.

Full text: PDF file (2978 kB)

English version:
Theory of Probability and its Applications, 1968, 13:1, 15–38

Bibliographic databases:

Citation: R. Sh. Liptser, A. N. Shiryaev, “Extrapolation of multidimensional Markov processes from incomplete data”, Teor. Veroyatnost. i Primenen., 13:1 (1968), 17–38; Theory Probab. Appl., 13:1 (1968), 15–38

Citation in format AMSBIB
\Bibitem{LipShi68} \by R.~Sh.~Liptser, A.~N.~Shiryaev \paper Extrapolation of multidimensional Markov processes from incomplete data \jour Teor. Veroyatnost. i Primenen. \yr 1968 \vol 13 \issue 1 \pages 17--38 \mathnet{http://mi.mathnet.ru/tvp790} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=235614} \zmath{https://zbmath.org/?q=an:0245.60038} \transl \jour Theory Probab. Appl. \yr 1968 \vol 13 \issue 1 \pages 15--38 \crossref{https://doi.org/10.1137/1113002}