In the book An Introduction to the Theory of Point Processes I by Vere-Jones exercise 7.2.5 asks to show that the intensity of a Hawkes process with exponential intensity kernel is Markov. I found various authors stating this property but rarely attempting to prove it or just refering to the memorylessness of the exponential. I would like to prove it with elementary methods and obtained the following $ Y(t) = \int_0^t \mathrm{exp}(-\beta(t-s))dN(s)$ $=\int_{t'}^t \mathrm{exp}(-\beta(t-s))dN(s) + \mathrm{exp}(-\beta(t-t'))\int_0^{t'} \mathrm{exp}(-\beta(t'-s))dN(s)$ $=\int_{t'}^t \mathrm{exp}(-\beta(t-s))dN(s) + \mathrm{exp}(-\beta(t-t'))Y(t')$

But how do I proceed from here? One idea was to use for $A\in\mathbb{R}^+$ with $A-x=\{a+x| a\in A\}$ for $x\in \mathbb{R}$ that

$\mathbb{P}[Y(t)\in A | Y(t')] = \mathbb{P}\left[\int_{t'}^t \mathrm{exp}(-\beta(t-s))dN(s) + \mathrm{exp}(-\beta(t-t'))Y(t')\in A |Y(t')\right]$ $=\mathbb{P}\left[\int_{t'}^t \mathrm{exp}(-\beta(t-s))dN(s)\in A-\mathrm{exp}(-\beta(t-t'))Y(t')|Y(t')\right]$

But then I am stuck, because I do not know how to introduce the whole natural filtration $\mathcal{F}_{t'}^Y$ to show the Markov property. Is there maybe a better way to do this?


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