Markovian Projection of Stochastic Processes
This PhD thesis studies various mathematical aspects of problems related to the Markovian projection of stochastic processes, and explores some ap- plications of the results obtained to mathematical finance, in the context of semimartingale models. Given a stochastic process ξ, modeled as a semimartingale, our aim is to build a Markov process X whose marginal laws are the same as ξ. This construction allows us to use analytical tools such as integro-differential equa- tions to explore or compute quantities involving the marginal laws of ξ, even when ξ is not Markovian. We present a systematic study of this problem from probabilistic view- point and from the analytical viewpoint. On the probabilistic side, given a discontinuous semimartingale we give an explicit construction of a Markov process X which mimics the marginal distributions of ξ, as the solution of a martingale problems for a certain integro-differential operator. This con- struction extends the approach of Gy ̈ongy to the discontinuous case and applies to a wide range of examples which arise in applications, in particu- lar in mathematical finance. On the analytical side, we show that the flow of marginal distributions of a discontinuous semimartingale is the solution of an integro-differential equation, which extends the Kolmogorov forward equation to a non-Markovian setting. As an application, we derive a forward equation for option prices in a pricing model described by a discontinuous semimartingale. This forward equation generalizes the Dupire equation, orig- inally derived in the case of diffusion models, to the case of a discontinuous semimartingale. These results give an application to the evaluation of index options allowing to reduce the problem of high dimension.