Bounds for the expected supremum of some non-stationary Gaussian processes

Abstract

The aim of this article is to obtain closed-form lower and upper bounds for the expectation of the supremum of some non-stationary Gaussian processes. This supremum is an important quantity in applications such as finance and queueing systems. The covariance function structure, decomposition of the processes and probability inequalities are applied for some examples of non-stationary Gaussian processes, including sub-fractional, bifractional and multifractional Brownian motions. Malliavin calculus operators are also applied in some cases to find bounds for the density and expectation of the supremum of these processes.

Keywords:

• Non-stationary Gaussian process
• Malliavin calculus
• subfractional Brownian motion
• bifractional Brownian motion
• multifractional Brownian motion

Acknowledgments

The authors would like to acknowledge that this work is supported, in part, by the Natural Sciences and Engineering Research Council of Canada (NSERC) through a Discovery Research Grant, and Carleton University. We also thank an anonymous reviewer and the editor for their valuable comments and suggestions on early versions of this work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors would like to acknowledge that this work is supported, in part, by the Natural Sciences and Engineering Research Council of Canada (NSERC) through a Discovery Research Grant, and Carleton University.