The conditional law of the Bacry–Muzy and Riemann–Liouville log correlated Gaussian fields and their GMC, via Gaussian Hilbert and fractional Sobolev spaces

Abstract

We compute E(Xt|(Xs)0≤s≤L) for the standard Bacry–Muzy log-correlated Gaussian field X with covariance log+Tt−s, which corrects the finite-horizon prediction formula in Vargas et al. (Duchon et al., 0000). The problem can be viewed as a linear filtering problem, and we solve the problem by showing that the L2(P) closure of {∫[0,L]ϕ(s)Xsds:ϕ∈S,supp(ϕ)⊆[0,L]} is equal to {X(ϕ):ϕ∈H−12,supp(ϕ)⊆[0,L]}, where X(ϕ) is defined as a continuous linear extension of X acting on S⊂Hs, Hs denotes the fractional Sobolev space of order s and P is the law of the field X on the space of tempered distributions. The explicit formula for the filter is obtained as the solution to a Fredholm integral equation of the first kind with logarithmic kernel. From this we characterize the conditional law of the Gaussian multiplicative chaos (GMC) Mγ generated by X, using that Mγ is measurable with respect to X. We also outline how one can adapt this result for the Riemann–Liouville GMC introduced in Forde et al. (2019), which has a natural application to the Rough Bergomi volatility model in the H→0 limit.11We would like to thank Juhan Aru, Janne Junilla, Vincent Vargas and Lauri Vitasaari for helpful discussions.