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For a Gaussian process $(Y_t)_{t\geq 0}$ with Hölder continuous paths, define a new process $(X_t^\varepsilon)_{t \geq 0}$ via

$$X_t^\varepsilon = \int_0^t \operatorname{sign}(Y_{s+\varepsilon}-Y_s) \, ds.$$

I would like to understand the limiting behaviour of $X_t^\varepsilon$ as $\varepsilon \to 0$ (a limit in distribution would be enough). In my concrete situation, $Y_t$ is fractional Brownian motion with a smooth perturbation, but I think that one can get rid of the perturbation by dividing the difference by $\varepsilon$ so that for small $\epsilon$ the fractional Brownian motion determines the sign. I tried decomposing the sign as

$$ \operatorname{sign}(Y_{s+\varepsilon}-Y_s) = 1_{\{ Y_{s+\varepsilon} - Y_s > 0\}} - 1_{\{ Y_{s+\varepsilon} - Y_s < 0\}} = 1_{\{Y_{s+\varepsilon} = Y_s\}} - 2 \times 1_{\{ Y_{s+\varepsilon} - Y_s < 0\}}, $$

so that the problem reduces to self intersecting local times and level/excursion sets but don't know how to go to the limit.

Does anybody know how to calculate such limits or can suggest some literature where problems of this kind are treated?

Thank you very much!

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  • $\begingroup$ Take a look at the proof for Tanaka's formula and related literature. $\endgroup$ – Thomas Kojar Nov 17 '17 at 23:25

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