Inference for diffusions driven by fractional Brownian motion
Résumé
First we state the almost sure convergence for the $k$-power second order increments of the fractional Brownian motion (fBm). Then we give the rate of that convergence, that is convergence in law for the $k$-power variation and more generally for a $g$ functional variation of the fBm, function $g$ being general but centered, including the absolute $k$-power variation. This allows us to propose several estimators of the Hurst parameter $H$ for the fBm, through the observation of one trajectory on a regular grid of points, using classical linear regression. The first one, $\widehat{H}_{k}$, is built using function $g_{k}(x)=\vert {x} \vert ^{k}- E\vert N \vert^{k}$, and the second one, $\widehat{H}_{\log}$, using function $g_{\log}(x) =\log\vert x\vert-E \log \vert N \vert$, leading to unbiased consistent estimators. A Central Limit Theorem (CLT) is also proposed for both estimators. The estimators $\widehat{H}_{k}$ and $\widehat{H}_{\log}$ are linked. That is, if $k(n)$ is a sequence of positive numbers converging to zero with $n$, and if $\widehat{H}_{k(n)}$ denotes the corresponding estimator of the $H$ parameter, we set out that the asymptotic behavior of $\widehat{H}_{k(n)}$ and of $\widehat{H}_{\log}$ are the same. The technics used before lead us to provide simultaneous estimators of parameter $H$ and of the local variance $\sigma$, in four particular simple models all driven by a fBm. As before, a regression model can be written and least square estimators of $H$ and of $\sigma$ are defined. These estimators are still built on the second order increments of the stochastic process solution of the stochastic differential equation. We still enunciate their consistency and a CLT is given for both of them. Furthermore, we consider testing the hypothesis $\sigma_{n}=\sigma$ against an alternative in the four previous models. An evaluation of the asymptotic power of the test is made. Finally simulations of the previous results are proposed.