It is common knowledge among schoolchildren that one may define jump diffusion processes in wide generality.

<b>Hard question:</b>  What are the most general structures on which one may define something which is appropriately called a jump diffusion process?

<b>Easier question:</b>  Can you define jump diffusions on Polish spaces?

I would love to see answers from both the <a href="https://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical-p/20828#20828">probabilistic</a> and <a href="https://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical-p/20820#20820">structuralist</a> perspectives.