Inference on the Hurst Parameter and the Variance of Diffusions Driven by Fractional Brownian Motion pp 109-122 | Cite as

# Complementary Results

## Abstract

In this chapter, we prove seven lemmas required in the detailed proofs of the results. The first two are related to the functional estimation seen in Sect. 5.3.3. Indeed, in that section we explain that in the case where *μ* ≡ 0, the solution of the SDE is *X*(*t*) = *K*(*b* _{ H }(*t*)) where *K* is a solution of an ODE and thus we assert that proving results enunciated in Remarks 3.28 and 3.30 is equivalent to prove them for the fBm. These lemmas give in an explicit manner how the increments of *X* can be approximated by those of the fBm. The proofs required the use of the modulus of continuity for the fBm and other results proved in Sect. 5.2.1. The third lemma is a straightforward calculation of the asymptotic variance of the random variable defined as a linear combination of variables of the type \(S_{g,\ell_{i}n}(1)\), used in Sect. 5.2.2. The fourth lemma is concerned by Sect. 5.2.3 where we link \(\hat{H}_{k(n)}\) with \(\hat{H}_{\log }\). In this lemma we proved that the corresponding functionals are equivalent in *L* ^{2}. For this aim we show that the Hermite coefficients for function \(\frac{g_{k(n)}} {k(n)}\) converge to those of function *g* _{log}. In the fifth lemma, we prove the almost sure equivalence between the second order increments of *X* and of *σ* times the increments of the fBm, referred to in Sect. 5.3.1. Giving the explicit solution for each of the four models and using the modulus of continuity for the fBm lead to the proof. A similar lemma is then demonstrated in the case where we do hypotheses testing seen in Sect. 5.3.2 replacing *σ* by *σ* _{ n } and the techniques are the same that for previous lemma. Finally in last and seventh lemma, we get back to functional estimation seen in Sect. 5.3.3 where *μ* is supposed to be null and where we want to prove the stable convergence for a functional of the fBm. This lemma is a step in this progression. More precisely, we prove the *L* ^{2} equivalence between the looked for functional and its approximation. That is done using regression techniques and straightforward calculus of expectations.