On Markovian projections for Itô semimartingales with jumps

<p dir="ltr">Given a general Itô semimartingale, its Markovian projection is an Itô process, with Markovian differential characteristics, that matches the one-dimensional marginal laws of the original process. One may even require certain functionals of the two processes to have the same fixed-time marginals, at the cost of enhancing the differential characteristics of the mimicking process but still in a Markovian sense. In applications, Markovian projections are useful in calibrating jump-diffusion models with both local and stochastic features, leading to the study of the inversion problems. In this thesis, we construct Markovian projections for Itô semimartingales with jumps via two different approaches. Apart from mimicking the process itself, in the second approach our mimicking theorem also works for a class of functionals of Itô semimartingales called updating functions. Lastly, we invert the Markovian projections for pure jump processes, which can be used to construct calibrated local stochastic intensity (LSI) models for credit risk applications. Such models are jump process analogues of the notoriously hard to construct calibrated local stochastic volatility (LSV) models used in equity modeling.</p>