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Gaussian and non-Gaussian processes of zero power variation
Probability · Submitted 4 Dec 2009
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Under some technical conditions on the quadratic variation of $M$, it is shown that the $m$-power variation exists and is zero when a quantity $\delta^{2}(r) $ related to the variance of an increment of $M$ over a small interval of length $r$ satisfies $\delta(r) = o(r^{1/(2m)}) $.
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This paper considers the class of stochastic processes $X$ which are Volterra convolutions of a martingale $M$. When $M$ is Brownian motion, $X$ is Gaussian, and the class includes fractional Brownian motion and other Gaussian processes with or without homogeneous increments. Let $m$ be an odd integer. Under some technical conditions on the quadratic variation of $M$, it is shown that the $m$-power variation exists and is zero when a quantity $\delta^{2}(r) $ related to the variance of an increment of $M$ over a small interval of length $r$ satisfies $\delta(r) = o(r^{1/(2m)}) $. In the case of a Gaussian process with homogeneous increments, $\delta$ is $X$'s canonical metric and the condition on $\delta$ is proved to be necessary, and the zero variation result is extended to non-integer symmetric powers. In the non-homogeneous Gaussian case, when $m=3$, the symmetric (generalized Stratonovich) integral is defined, proved to exist, and its It\^o's formula is proved to hold for all functions of class $C^{6}$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This paper considers the class of stochastic processes $X$ which are Volterra convolutions of a martingale $M$. | contribution 0.79 | 0.58 | 0.04 | asserted 0.96 |
| s1 | When $M$ is Brownian motion, $X$ is Gaussian, and the class includes fractional Brownian motion and other Gaussian processes with or without homogeneous increments. | background 0.70 | 0.30 | 0.06 | asserted 1.00 |
| s2 | Let $m$ be an odd integer. | method 0.57 | 0.22 | 0.01 | asserted 0.55 |
| s3 | Under some technical conditions on the quadratic variation of $M$, it is shown that the $m$-power variation exists and is zero when a quantity $\delta^{2}(r) $ related to the variance of an increment of $M$ over a small interval of length $r$ satisfies $\delta(r) = o(r^{1/(2m)}) $. | result 0.99 | 0.93 | 0.57 | asserted 1.00 |
| s4 | In the case of a Gaussian process with homogeneous increments, $\delta$ is $X$'s canonical metric and the condition on $\delta$ is proved to be necessary, and the zero variation result is extended to non-integer symmetric powers. | result 0.84 | 0.91 | 0.19 | asserted 1.00 |
| s5 | In the non-homogeneous Gaussian case, when $m=3$, the symmetric (generalized Stratonovich) integral is defined, proved to exist, and its It\^o's formula is proved to hold for all functions of class $C^{6}$. | contribution 0.53 | 0.93 | 0.13 | asserted 1.00 |
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Original abstract
This paper considers the class of stochastic processes $X$ which are Volterra convolutions of a martingale $M$. When $M$ is Brownian motion, $X$ is Gaussian, and the class includes fractional Brownian motion and other Gaussian processes with or without homogeneous increments. Let $m$ be an odd integer. Under some technical conditions on the quadratic variation of $M$, it is shown that the $m$-power variation exists and is zero when a quantity $\delta^{2}(r) $ related to the variance of an increment of $M$ over a small interval of length $r$ satisfies $\delta(r) = o(r^{1/(2m)}) $. In the case of a Gaussian process with homogeneous increments, $\delta$ is $X$'s canonical metric and the condition on $\delta$ is proved to be necessary, and the zero variation result is extended to non-integer symmetric powers. In the non-homogeneous Gaussian case, when $m=3$, the symmetric (generalized Stratonovich) integral is defined, proved to exist, and its It\^o's formula is proved to hold for all functions of class $C^{6}$.
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States the same result 1 paper
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Under the assumption that the quadratic variation $\left[ M\right] $ of $M$ is differentiable with $\mathbf{E}\left[ \left\vert d\left[ M\right] (t)/dt\right\vert ^{m}\right] $ finite, it is shown that the $m$th power variation $$ \lim_{\varepsilon\rightarrow0}\varepsilon^{-1}\int_{0}^{T}ds\left( X\left( s+\varepsilon\right) -X\left( s\right) \right) ^{m} $$ exists and is zero when a quantity $\delta^{2}\left( r\right) $ related to the variance of an increment of $M$ over a small interval of length $r$ satisfies $\delta\left( r\right) =o\left( r^{1/(2m)}\right) $.
Gaussian and non-Gaussian processes of zero power variation, and related stochastic calculus · 0.89
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Compares with 1 paper
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The paper [ref] does not follow, however, the same motivation as our work: for us, say in the case of $m=3$, the threshold $H>1/6$ is the key point for (odd) cubic variations, with its relation to stochastic calculus.
Fractional martingales and characterization of the fractional Brownian motion · Relation with other recent work · 0.99
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