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Gatheral (Amazon) has a quite extensive discussion on that, and dives into calibration issues. In summary, what you describe appears to be less of a modeling issue, and more of a calibration problem. This is primarily because the model functions (such as the Heston model) are not by nature convex in their input parameters. This is simply result of the fact ...


don't know If I understand well your question, but If you want to have a rather complete perspective about the affine class of models (to which Heston's model belongs), you better study Duffie et al. (2000). In this very important contribution you'll find many examples of jump specifications


Yes! Try this and this. But if you don't know the black-scholes basics well consider to read the book "Paul Wilmott in Quantitative Finance" before to go to Stochastic Volatility models and models with jumps.

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