Mixed Gaussian processes: A filtering approach

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 \[K(s,t)=\frac{\partial^{2}}{\partial s\,\partial t}\mathbb{E}G_{t}G_{s},\qquad s\ne t\] 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.

Cite this work

No public copy is stored for this record, so only the metadata is cited.

Stored only when both papers are in the corpus — 53 citations exist upstream.

Papers that cite the same references — the relational fallback while embeddings are still backfilling.