Prediction law of mixed Gaussian Volterra processes

Abstract

We study the regular conditional law of mixed Gaussian Volterra processes under the influence of model disturbances. More precisely, we study prediction of Gaussian Volterra processes driven by a Brownian motion in a case where the Brownian motion is not observable, but only a noisy version is observed. As an application, we discuss how our result can be applied to variance reduction in the presence of measurement errors.

Long-range dependent completely correlated mixed fractional Brownian motion

“…Naturally, we are interested in predicting the future, i.e., we are interested in the conditional future probability law of the process X given the information F X u . The transfer principle of Theorem 2 provides us these prediction formulas for the ccmfBm in the same way as in [21,22]:…”

Pathwise asymptotics for Volterra processes conditioned to a noisy version of the Brownian motion

“…The following theorem, Theorem 2.1 in [16], gives mean and covariance function of the Gaussian conditional law. Theorem 5.…”

“…In this section (X t ) t≥0 is a continuous Volterra process as in (1). Now, in order to achieve a large deviation principle for the generalized conditioned process X ψ , we have to investigate the behavior of the functions Υ and m ψ (defined in (16) and (17), respectively) in a small time interval of length ε. We want to investigate the behavior of the conditioned process (X ψ t ) t≥0 in the near future after T .…”