Information loss on Gaussian Volterra process. (English) Zbl 1386.60142

Valdivia, Arturo

Electron. Commun. Probab. 22, Paper No. 60, 5 p. (2017).

Summary: Gaussian Volterra processes are processes of the form \((X_{t}:=\int_{\mathbf{T}}k(t,s)\text{d}W_{s})_{t\in \mathbf{T}}\) where \((W_{t})_{t\in \mathbf{T}}\) is Brownian motion, and \(k\) is a deterministic Volterra kernel. On integrating the kernel \(k\) an information loss may occur, in the sense that the filtration of the Volterra process needs to be enlarged in order to recover the filtration of the driving Brownian motion. In this note we describe such enlargement of filtrations in terms of the Volterra kernel. For kernels of the form \(k(t,s)=k(t-s)\) we provide a simple criterion to ensure that the aforementioned filtrations coincide.

MSC:

60G15	Gaussian processes
60G22	Fractional processes, including fractional Brownian motion
60H20	Stochastic integral equations
60J65	Brownian motion
91G99	Actuarial science and mathematical finance

Keywords:

enlargement of filtrations; long range dependence; superposition of Ornstein-Uhlenbeck processes; Volterra process