Mixed Gaussian processes: A filtering approach

TL;DR

This paper introduces a novel filtering-based approach to analyze mixed Gaussian processes, generalizing classical results and applying to processes with fractional structures, relevant in finance and stochastic analysis.

Contribution

It develops a new canonical innovation representation for mixed Gaussian processes, extending classical formulas to cases with singular kernels and fractional structures.

Findings

Unified representation for mixed Gaussian processes

Generalization of innovation formulas beyond square integrability

New Radon-Nikodym density formulas for measure changes

Abstract

This paper presents a new approach to the analysis of mixed processes \[X_t=B_t+G_t,\qquad t\in[0,T],\] where $B_t$ is a Brownian motion and $G_t$ is an independent centered Gaussian process. We obtain a new canonical innovation representation of $X$, using linear filtering theory. When the kernel \[K(s,t)=\frac{\partial^2}{\partial s\,\partial t}\mathbb{E}G_tG_s,\qquad s\ne t\] has a weak singularity on the diagonal, our results generalize the classical innovation formulas beyond the square integrable setting. For kernels with stronger singularity, our approach is applicable to processes with additional "fractional" structure, including the mixed fractional Brownian motion from mathematical finance. We show how previously-known measure equivalence relations and semimartingale properties follow from our canonical representation in a unified way, and complement them with new formulas for Radon-Nikodym densities.