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associativity of stochastic integration
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(Theorem)
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The chain rule for expressing the derivative of a variable $z$ with respect to $x$ in terms of a third variable $y$ is \begin{equation*} \frac{dz}{dx}=\frac{dz}{dy}\frac{dy}{dx}. \end{equation*}Equivalently, if $dy=\alpha\,dx$ and $dz=\beta\,dy$ then $dz=\beta\alpha\,dx$ . The following theorem shows that the stochastic integral satisfies a generalization of this.
Theorem Let $X$ be a semimartingale and $\alpha$ be an $X$ -integrable process. Setting $Y=\int\alpha\,dX$ then $Y$ is a semimartingale. Furthermore, a predictable process $\beta$ is $Y$ -integrable if and only if $\beta\alpha$ is $X$ -integrable, in which case \begin{equation}\label{eq:1} \int\beta\,dY=\int\beta\alpha\,dX. \end{equation}
Note that expressed in alternative notation, (![[*]](http://images.planetmath.org:8080/cache/objects/11436/js//usr/share/latex2html/icons/crossref.png "[*]") ) becomes \begin{equation*} \beta\cdot(\alpha\cdot X)=(\beta\alpha)\cdot X \end{equation*}or, in differential notional, \begin{equation*} \beta(\alpha\,dX)=(\beta\alpha)\,dX. \end{equation*}That is, stochastic integration is associative.
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"associativity of stochastic integration" is owned by gel.
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stochastic integral, semimartingale |
This object's parent.
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Cross-references: associative, predictable process, semimartingale, stochastic integral, theorem, variable, chain rule
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This is version 1 of associativity of stochastic integration, born on 2009-01-01.
Object id is 11436, canonical name is AssociativityOfStochasticIntegration.
Accessed 366 times total.
Classification:
| AMS MSC: | 60H05 (Probability theory and stochastic processes :: Stochastic analysis :: Stochastic integrals) | | | 60G07 (Probability theory and stochastic processes :: Stochastic processes :: General theory of processes) | | | 60H10 (Probability theory and stochastic processes :: Stochastic analysis :: Stochastic ordinary differential equations) |
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Pending Errata and Addenda
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