# Markovian Hawkes Process elementary proof

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?