Approximation at First and Second Order of $m$-order Integrals of the Fractional Brownian Motion and of Certain Semimartingales

By Mihai Gradinaru, Ivan Nourdin. 2003

54 citations recorded by the source corpus.

Let $X$ be the fractional Brownian motion of any Hurst index $H\in (0,1)$ (resp. a semimartingale) and set $\alpha=H$ (resp. $\alpha=\frac{1}{2}$). If $Y$ is a continuous process and if $m$ is a positive integer, we study the existence of the limit, as $\varepsilon\rightarrow 0$, of the approximations $$ I_{\varepsilon}(Y,X) :=\left\{\int_{0}^{t}Y_{s}\left(\frac{X_{s+\varepsilon}-X_{s}}{\varepsilon^{\alpha}}\right)^{m}ds,\,t\geq 0\right\} $$ of $m$-order integral of $Y$ with respect to $X$. For these two choices of $X$, we prove that the limits are almost sure, uniformly on each compact interval, and are in terms of the $m$-th moment of the Gaussian standard random variable. In particular, if $m$ is an odd integer, the limit equals to zero. In this case, the convergence in distribution, as $\varepsilon\rightarrow 0$, of $\varepsilon^{-\frac{1}{2}} I_{\varepsilon}(1,X)$ is studied. We prove that the limit is a Brownian motion when $X$ is the fractional Brownian motion of index $H\in (0,\frac{1}{2}]$, and it is in term of a two dimensional standard Brownian motion when $X$ is a semimartingale.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2652cae9e803ba4bf5ac6e4e8958276b47ac63b4

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Citations in the corpus

Itô's formula for a sub-fractional Brownian motion (2011, background): “By a proof similar to Lemma 3.1 in Gradinaru–Nourdin [14], one can obtain the following convergence (see also Gradinaru–Nourdin [15]): lim n→∞ n∑ j=1 j2H−1g(SHtj )(S H tj − S H tj−1) 2 = ∫ t 0 g(SHs )s 2H−1ds (6.12) almost surely, where tj = tj/n and g ∈ C(R).”
ASYMPTOTIC THEORY FOR ESTIMATING DRIFT PARAMETERS IN THE FRACTIONAL VASICEK MODEL (2018, background): “One possibility for estimating H is to use the generalized quadratic variation introduced by Gradinaru and Nourdin (2006). For T > 0, β > 0, γ > 0 and β = γ , assume Xt is observed continuously over the interval [0,T + max(β,γ )].”
Elements of Stochastic Calculus via Regularization (2006, background): “In [25], it is shown that stochastic calculus via regularization is almost pathwise.”