Double Telegraph Processes and Complete Market Models

Abstract

The traditional jump-telegraph processes are based on a Poisson process with alternating intensities. We develop a new model based on an alternating doubly stochastic Poisson process with random intensities of jumps. Moreover, in this model the jump-telegraph process performs additional jumps each time when the intensity changes at random. Martingale measures for this type of processes are described by using Girsanov’s transformation. The market model based on the doubly stochastic jump-telegraph process is studied. A class of complete models is also considered.

Keywords:

• Markov flow
• Doubly stochastic Poisson process
• Martingale
• Jump-telegraph process
• Complete market models

Mathematics Subject Classification:

• 60J27
• 60J75
• 60K99
• 91G99

Acknowledgments

The author thank the anonymous referee for helpful comments.