Financial Markets with Memory II: Innovation Processes and Expected Utility Maximization

Abstract

We develop a prediction theory for a class of processes with stationary increments. In particular, we prove a prediction formula for these processes from a finite segment of the past. Using the formula, we prove an explicit representation of the innovation processes associated with the stationary increments processes. We apply the representation to obtain a closed-form solution to the problem of expected logarithmic utility maximization for the financial markets with memory introduced by the first and second authors.

Key Words:

• Financial market model
• Innovation process
• Utility maximization

Mathematics Subject Classification:

• Primary 91B28
• Secondary 60G25, 62M20

ACKNOWLEDGMENTS

This work is partially supported by the Australian Research Council grant A10024117 and Grant–in–Aid for JSPS Fellows 14007206.