Martingales on Jump Processes. I: Representation Results

Rene K. Boel · Pravin P. Varaiya

149 citations0/0 files storedSIAM Journal on Control

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Stochastic processes and financial applications Probability and Statistical Research Probability and Risk Models

Abstract

The paper is a contribution to the theory of martingales of processes whose sample paths are piecewise constant and have finitely many discontinuities in a finite time interval. The assumption is made that the jump times of the underlying process are totally inaccessible and necessary and sufficient conditions are given for this to be true. It turns out that all martingales are then discontinuous, and can be represented as stochastic integrals of certain basic martingales. This representation theorem is used in a companion paper to study various practical problems in communication and control. The results in the two papers constitute a sweeping generalization of recent work on Poisson processes.

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References

1960On Transforming a Certain Class of Stochastic Processes by Absolutely Continuous Substitution of Measures 1967On Square Integrable Martingales 1971Existence of Optimal Stochastic Control Laws 1970The Representation of Functionals of Brownian Motion by Stochastic Integrals 1973Dynamic Programming Conditions for Partially Observable Stochastic Systems 1970Quelques applications de la formule de changement de variables pour les semimartingales

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1976The Representation of Martingales of Jump Processes 1977Optimal Control of Jump Processes 1978Modeling and analysis of stochastic differential equations driven by point processes 1999An Equilibrium Model of Catastrophe Insurance Futures and Spreads 2005Martingale methods in stochastic control 1977Weak convergence of stochastic integrals related to counting processes 1987A review on stochastic differential equations for applications in hydrology 2010

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