Differentiating σ-fields for Gaussian and shifted Gaussian processes

Abstract

We study the notions of differentiating and non-differentiating σ-fields in the general framework of (possibly drifted) Gaussian processes and characterize their invariance properties, when changing to an equivalent probability measure. As an application, we investigate the class of stochastic derivatives associated with shifted fractional Brownian motions. We finally establish conditions for the existence of a jointly measurable version of the differentiated process and we outline a general framework for stochastic embedded equations.

Keywords:

• differentiating σ-fields
• stochastic derivatives
• Nelson's derivatives
• fractional Brownian motion
• fractional differential equations

AMS Subject Classification:

• Primary: 60G07, 60G15
• Secondary: 60G17, 60H07

Acknowledgement

We would like to thank the anonymous referee for a careful and thorough reading.

Notes

^1. [email protected]

^2. [email protected]

Additional information

Notes on contributors

Ivan Nourdin

1. 1. [email protected]

Giovanni Peccati

2. 2. [email protected]