Uniformly and strongly consistent estimation for the random Hurst function of a multifractional process

Multifractional processes are extensions of Fractional Brownian Motion obtained by replacing its constant Hurst parameter by a deterministic or a random function H(•), called the Hurst function, which allows to prescribe their local sample paths roughness at each point. For that reason statistical estimation of H(•) is an important issue. Many articles have dealt with this issue in the case where H(•) is deterministic. However, statistical estimation of H(•) when it is random remains an open problem. The main goal of our present article is to propose, under a weak local Hölder condition on H(•), a solution for this problem in the framework of Moving Average Multifractional Process with Random Exponent (MAMPRE), denoted by X. From the data consisting in a discrete realization of X on the interval [0, 1], we construct a continuous piecewise linear random function which almost surely converges to H(•) for the uniform norm, when the size of the discretization mesh goes to zero; also we provide an almost sure estimate of the uniform rate of convergence. It is worth noticing that such kind of strong consistency result in uniform norm is rather unusual in literature on statistical estimation of functions.