Inference on the Hurst Parameter and the Variance of Diffusions Driven by Fractional Brownian Motion

The chapter introduces the basic tools we use. First the fBm, its harmonizable representation and some of its properties are given, as well as an evaluation of its modulus of continuity. Also it is shown that fBm is not a semi-martingale. Then some preliminaries about stochastic integration with respect to the fBm are introduced. The attention will be focused on pathwise integrals. The existence of the integral as the limit in probability of the Riemann sums is proved. Afterwards, the notion of a stochastic differential equation driven by fractional noise is given, explaining next how one can construct the solution of such an equation. Moreover, it is established that the definitions of backwards, forwards and symmetric pathwise integrals coincide for some class of integrant functions. After the complex Itô-Wiener Chaos is defined and some tools useful for the subject are provided: Mehler’s formula, normalized second order increments and its covariance function. The chapter concludes by defining generalized variations.

The first theorem of this chapter establishes the almost sure convergence for the k-power second order increments of the fractional Brownian motion (fBm) toward the k-th moment of a standard normal distribution. Then we give the rate of this convergence in law. Moreover, for a general functional variation of the fBm, see (2.​4), page 40, including the absolute k-power variation, the result remains true. This allows us to propose several estimators of the Hurst parameter H for the fBm using classical linear regression. The first one, \(\hat{H}_{k}\), uses the function \(\left \vert x\right \vert ^{k}\), and the second one, \(\hat{H}_{\log }\), uses the Napierian logarithm and both lead to unbiased consistent estimators. A Central Limit Theorem (CLT) is also obtained for both estimators. These estimators are linked in the sense that if k(n) is a sequence of positive numbers converging to zero with \(n\), and if \(\hat{H}_{k(n)}\) denotes the corresponding estimator of the H parameter, we set out that the asymptotic behaviors of \(\hat{H}_{k(n)}\) and of \(\hat{H}_{\log }\) are the same. The same techniques can be used to provide simultaneous estimators of parameter H and of the local variance σ, in four particular simple models all driven by a fBm. As before, a regression model can be written and least squares estimators of H and of σ are defined. These estimators are built on the second order increments of the stochastic process solution of the proposed model. We prove their consistency and a CLT is given for both of them. Furthermore, we consider testing the hypothesis σ _n  = σ against an alternative in the four previous models. Finally, we propose functional estimation of the local variance of general stochastic differential equation (SDE). This estimation is based on the observation of the second order increments of the solution of such an SDE. We highlight that to show the convergence in these models, it is sufficient to prove it in the special case where the solution process is the fBm.

In this chapter, we present the basic ideas for the simulation of a stationary Gaussian process from which we deduce the simulation of a fBm and the simulation of processes driven by a fBm. Our approach is based on Durbin-Levinson’s algorithm. Since the process formed by the first order increments of a fBm is a stationary Gaussian one, we first simulate the increments of the process and then, by a simple “integration”, we obtain a trajectory of the fBm. For models defined by differential equations, first an observation of the fBm is generated and then, it is transformed according to differential equation. Simulating these processes, we can explore the statistical properties of the estimators defined in the previous chapter from an empirical point of view. We study the distribution of the estimators of H and of σ. Special attention is devoted to the construction of a confidence interval for H. Some simulation results concern the estimation of the parameters of a pure fBm, some others are for the parameters of models that are excited by a fBm. To generate the uniform deviates, we recommend the use of linear congruential generator given in Langlands et al. (Bull Am Math Soc (New Ser) 30(1):1–61, 1994) and for the normal deviates, and we recommend the use of Algorithm M described in Knuth (The art of computer programming. Vol. 2. Seminumerical algorithms, Addison-Wesley series in computer science and information processing, 2nd edn. Addison-Wesley, Reading, 1981). It is a very fast generator. Pascal programs are given in Chap. 8.

In this chapter dedicated to the proofs of the various results, we explore the properties of three kinds of estimators for the Hurst parameter of the fBm. These estimators are built on the second order increments of fBm that allows estimation all over the range of parameter H in \(\left ]\,0,\ 1\,\right [\,\). We prove a CLT for simultaneous estimators of the Hurst parameter H and of the local variance σ in the four following models: \(\mathrm{d}X(t) =\sigma (X(t))\mathrm{d}b_{H}(t) +\mu (X(t))\mathrm{d}t\), where σ(x) = σ or σ x and μ(x) = μ or μ x. When H is supposed to be known, test of hypotheses on σ are proposed. Finally, functional estimation is considered for function σ in the following model: \(\mathrm{d}X(t) =\sigma (X(t))\mathrm{d}b_{H}(t) +\mu (X(t))\mathrm{d}t\), where functions σ and μ verify technical hypotheses. In this chapter we used the techniques of the CLT for functionals that belong to Wiener chaos and more precisely the one of the Peccati-Tudor theorem.

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.

In this chapter we collect all the graphics and tables to which we refer in the text. The are presented to help the understanding of the different comments concerning the simulations results. First, we display three graphics showing the empirical distribution of \(\hat{H}_{2}\) obtained with a resolution of 1/2,048-th for different values of H. Then, in Tables 7.1–7.5, we give the empirical mean and standard deviation of the estimators of H in the case of a fBm. Graphical representations are presented on pages 130–131. Tables 7.6 and 7.7 present some results concerning the estimated covering probability of the confidence intervals we developed in Sects. 4.​5.​1.​3 and 4.​5.​1.​4, pages 67 and 70. A series of Tables 7.8–7.15, followed by a series of graphics, Figs. 7.4–7.11 present results about the simultaneous estimation of H and σ for models excited by an fBm. Table 7.16 gives the observed empirical level of the test on σ. Figures 7.12–7.19 present the empirical and the asymptotic power function of the test.

In this chapter, we give the important Pascal procedures used in the simulation studies: the uniform and the normal generators. These are the basic functions use in the procedure DurbinSim written to simulate a trajectory of a Gaussian stationary process. If the increments are simulated, the function Somme is used to get the trajectory. We also give the procedure Model that control the simulation of the four different models defined by a stochastic differential equation considered in the text.