# 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.

• I don't have the time for a full answer at the moment. Have a look at Proposition 3.6 (p. 78) and Proposition 3.17 (p. 97) in Cont and Tankov "Financial Modelling with Jump Processes". See also the more general Proposition 19.5 (p. 123) in Sato "Levy Processes an Infinitely Divisible Distributions". – LocalVolatility Jun 5 '18 at 8:50