Why should the expectation of the quadratic variation be finite? I thought this is true if and only if the martingale is square integrable?

While I agree that concepts of convergence can be understood only via measure theory, one can get to a certain degree a good understanding of limit theorems without it. E.g., Emanuel Lesigne's "Heads or Tails" (ams.org/bookstore-getitem/item=STML-28) is a very readable introduction.

I agree with this, only that the process can become numerically negative even in the case that Feller's condition is satisfied. An alternative way to circumvent this (due to Alfonsi, cermics.enpc.fr/~alfonsi/SC_preprint.pdf) is to rewrite the Ito Integrals into Stratonovich integrals and discretize only then. This leads to an implicit FD scheme that can even be turned into an explicit one if Feller's condition is satisfied.

I can confirm that when calibrating the Heston model to equity data, the parameter are usually close to the boundary, sometimes they satisfy Feller, sometimes they don't. But this is a problem only insofar, as some theoretical results on the model require the condition to hold and you have to be careful to check if a given paper requires the Feller condition or it doesn't.

And if you make this assumption, neither the diffusion nor the jump diffusion you specified reach zero a.s.

I still do not understand what you mean by "The Feller condition holds". It is just an assumption on parameters.

What do you mean by "need" the Feller condition. Do you have some specific proof in mind where as it is needed? the only thing it does is to assure you that you do not hit 0 a.s. As you have in your example only an upward jump part with finite jump actitvity, its presence does not change anything in this consideration.

I agree that he sweeps some things under the rug, one could add his treatment of quadratic variation or the non-treatment of local martingales (a quite important topic even for option pricing, e.g. in local and stochastic volatility models). But the balance between ease of exposition and rigor is hard to strike (do you really want to discuss tightness of probability measures in such a class?) and I think he is doing a very good job. I feel amending a textbook by other sources is now easier than ever, as many textbooks are in electronic form available for students for free...

I assume by moments you mean the moments of the marginals for some fixed t? I doubt that this will help much without knowing the transition probabilities of $X$. Could you be more precise how your process $X$ is given?

Your process $Z_t$ admits the explicit representation $e^{-\kappa t}\Bigl(Z_0 + \kappa \int_0^t e^{\kappa s} X_s \, ds\Bigr)$. So without knowing the distribution of $X_t$ or some particular properties (as bounded away from zero), I do not expect a general result.

A good overview of Doblin's work in the "pli chachete" is given in this article by Bernard Bru and Marc Yor: ams.org/mathscinet/search/…

I am not sure how much this can count as rediscovery. There is no stochastic integral in Doblin's paper, and hence no actual Ito formula. What Doblin does, is to give a probabilistic characterization of a diffusion in terms of a time-changed Brownian motion. On the other hand this can be considered as even more anticipating, as the connection of random time changes and stochastic integrals was only discovered in 1965, by Dubins-Schwarz and Dambis...

Related question: mathoverflow.net/questions/106808/…

While not an exact duplicate, does this answer your question? mathoverflow.net/questions/46957/…