The two-parameter Volterra multifractional process

In the case where the parameters H1 and H2 belong to (1/2,1), Feyel and De La Pradelle (1991) have introduced a representation of the usual fractional Brownian sheet {Bs,tH1,H2}(s,t)∈R+2, as a stochastic integral over the compact rectangle [0,s]×[0,t], with respect to the Brownian sheet. In this paper, we introduce the so-called two-parameter Volterra multifractional process by replacing in the latter representation of {Bs,tH1,H2}(s,t)∈R+2 the constant parameters H1 and H2 by two Hölder functions α(s) and β(t) with values in (1/2,1). We obtain that the pointwise and the local Hölder exponents of the two-parameter Volterra multifractional process at any point (s0,t0) are equal to min(α(s0),β(t0)).