Integral representation of generalized grey Brownian motion

Area of Science:

• Stochastic processes
• Mathematical finance
• Probability theory

Background:

• The Ornstein-Uhlenbeck process is a fundamental model in stochastic calculus.
• Existing research has established representation results for Gaussian processes.
• Generalizing these results to non-Gaussian processes remains an open area.

Purpose of the Study:

• To investigate the representation of generalized grey Brownian motion.
• To establish a connection between non-Gaussian processes and stochastic differential equations.
• To extend existing representation theorems to a broader class of processes.

Main Methods:

• Utilizing weighted integrals of stochastic processes.
• Analyzing solutions to specific stochastic differential equations.
• Developing a non-Gaussian extension of the Ornstein-Uhlenbeck process.

Main Results:

• A novel representation for generalized grey Brownian motion is derived.
• The findings generalize previously established results for Gaussian processes.
• The underlying process is shown to be a non-Gaussian extension of the Ornstein-Uhlenbeck process.

Conclusions:

• The paper provides a new framework for understanding non-Gaussian stochastic processes.
• This research bridges the gap between Gaussian and non-Gaussian process representations.
• The results have potential implications for fields utilizing stochastic modeling.