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

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

Easier question: Can you define jump diffusions on Polish spaces?

I would love to see answers from both the probabilistic and structuralist perspectives.

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

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

Easier question: Can you define jump diffusions on Polish spaces?

I would love to see answers from both the probabilistic and structuralist perspectives.