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STOCHASTIC INTEGRATION FOR TEMPERED FRACTIONAL BROWNIAN MOTION.
Mark M Meerschaert1, Farzad Sabzikar2
1D epartment of S tatistics and P robability , M ichigan S tate U niversity , E ast L ansing MI 48823 mcubed@stt.msu.edu URL: http://www.stt.msu.edu/users/mcubed/
Stochastic Processes and Their Applications
|May 30, 2014
Summary
This study introduces stochastic integrals for tempered fractional Brownian motion, a process modified by an exponential tempering factor. The research lays foundational theory for this advanced mathematical concept.
Area of Science:
- Stochastic calculus
- Fractional calculus
- Time series analysis
Background:
- Fractional Brownian motion (fBm) is a key model in stochastic processes.
- Modifications to fBm, such as tempering, are crucial for capturing complex real-world phenomena.
- Tempering fBm involves altering its kernel with an exponential factor.
Purpose of the Study:
- To develop the mathematical theory of stochastic integrals for tempered fractional Brownian motion (tfBm).
- To establish foundational results in the calculus of tfBm.
- To extend the analytical tools available for analyzing tempered stochastic processes.
Main Methods:
- Developing a novel framework for stochastic integration tailored to tfBm.
- Deriving key properties and theorems related to tfBm stochastic integrals.
- Establishing basic results in tempered fractional calculus.
Main Results:
- The successful development of stochastic integral theory for tfBm.
- New insights into the behavior of tfBm through its integral representations.
- Foundational results in tempered fractional calculus enabling further research.
Conclusions:
- The theory of stochastic integrals for tfBm is now established.
- This work provides essential tools for researchers in stochastic analysis and related fields.
- Further applications of tfBm in finance, physics, and other areas are now more accessible.
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