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 Teor. Veroyatnost. i Primenen., 1972, Volume 17, Issue 1, Pages 167–172 (Mi tvp4215)

Short Communications

Representations of Itô Processes

M. P. Ershov

Abstract: Let $(\Omega,\mathscr{F},\mathbf{P})$ be a complete probability space. By an Itô process relative to an increasing family $\{\mathscr{F}_t\}$ of sub-$\sigma$-algebras of $\mathscr{F}$, we mean a process $\xi$ of the form
$$\xi_t=\xi_0+\int_0^t\alpha_s ds+\int_0^t \beta_s dW_s$$
where $\alpha,\beta$ are measurable processes well adapted to $\{\mathscr{F}_t\}$, $\displaystyle\int_0^t (|\alpha_s|+\beta_{s}^2)ds<\infty$ $\forall t$ a.s., and $W$ is a standard Wiener process with respect to $\mathscr{F}$. We study conditions under which an Itô process $\xi$ relative to $\{\mathscr{F}_t\}$ is also an Itô process relative to a family $\{\mathscr{G}_t\}$ of “simpler” $\sigma$-algebras: $\mathscr{G}_t\subseteq\mathscr{F}_t$ for each $t$.

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English version:
Theory of Probability and its Applications, 1972, 17:1, 165–169

Bibliographic databases:

Citation: M. P. Ershov, “Representations of Itô Processes”, Teor. Veroyatnost. i Primenen., 17:1 (1972), 167–172; Theory Probab. Appl., 17:1 (1972), 165–169

Citation in format AMSBIB
\Bibitem{Ers72} \by M.~P.~Ershov \paper Representations of It\^o Processes \jour Teor. Veroyatnost. i Primenen. \yr 1972 \vol 17 \issue 1 \pages 167--172 \mathnet{http://mi.mathnet.ru/tvp4215} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=300339} \zmath{https://zbmath.org/?q=an:0301.60056} \transl \jour Theory Probab. Appl. \yr 1972 \vol 17 \issue 1 \pages 165--169 \crossref{https://doi.org/10.1137/1117016} 

• http://mi.mathnet.ru/eng/tvp4215
• http://mi.mathnet.ru/eng/tvp/v17/i1/p167

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. V. Selivanov, “On time changes for Lévy processes”, Russian Math. Surveys, 58:2 (2003), 388–389