On the Theory of (Dual) Projection for Fuzzy Stochastic Processes

Abstract

In a theory similar to one of real-valued stochastic processes, in this paper, we investigate the projection and dual projection for fuzzy stochastic processes. First, the related concepts of fuzzy stochastic processes are introduced, such as adaption, measurability, optionality, predictability, etc. Subsequently, we study fuzzy stochastic integral and fuzzy measure generated by increasing fuzzy stochastic processes. Moreover, (dual) projection w.r.t. (increasing) fuzzy stochastic processes are discussed. We prove the existence and uniqueness of (dual) optional (predictable) projection for (increasing) fuzzy stochastic processes.

Key Words:

• Dual optional (predictable) projection
• Fuzzy measure
• Fuzzy stochastic integral
• Increasing fuzzy stochastic processes
• Optional (predictable) projection.

Mathematics Subject Classification:

• Primary 03E72, 28E10
• Secondary 60G05, 60H05.

ACKNOWLEDGMENTS

The author would like to thank the referees for valuable comments and suggestions. Supported by the Key Project of Education Ministry of China (205073) and the National Natural Science Foundation of China (70071016).