Exponential Martingales for Inhomogeneous Poisson Processes

Summary

The document poses a question about proving that an exponential expression involving a counting process with time-varying intensity is a martingale. The expression combines the accumulated predictable process at event times with a compensating integral of the intensity and the exponential transform of that process. The question specifically asks how to establish the property for every bounded predictable process, and notes that a monotone class theorem is suggested as a proof technique.

No proof or answer is included, so the document does not develop the theorem or provide supporting calculations. Its value is mainly in identifying a martingale result and a likely route to proving it. A complete treatment would need to state the assumptions on the intensity and filtration, then justify the result first for a tractable class of predictable processes and extend it appropriately. It is a mathematical question rather than a trading strategy or empirical analysis.

Key ideas

• The document asks whether an exponential transform of a counting process with time-varying intensity is a martingale.
• The integrand is required to be bounded and predictable with respect to the counting process filtration.
• A monotone class theorem is suggested as a possible route to extending a proof.
• The document contains no derivation or stated assumptions beyond the question.

Tags

# Martingale property of inhomogenous poisson process


# Martingale property of inhomogenous poisson process












I have found this martingale property for an inhomogenous poisson process with intensity $\lambda(s)$ which I don't know how to prove. The text itself advises: "proceed using Monotone class theorem". Any idea how to show that for any bounded $\mathcal{F}^N$ predictable process $H_s$ this is a martingale?

$$\exp \left( \int_0^tH_sdN_s - \int_0^t \lambda(s) \left( e^{H_s} - 1 \right) ds \right)$$

Thanks for suggestions.

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