Projection markovienne de processus stochastiques

tel-00766235, version 1 - 17 Dec 2012
Curie- Paris VI) under the supervision of Rama Cont. It studies various
mathematical aspects of problems related to the Markovian projection of
stochastic processes, and explores some applications 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
constructionallowsustouseanalyticaltoolssuchasintegro-differentialequations
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 viewpoint
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 (Chapter 2).
This construction extends the approach of Gyöngy to the discontinous case
and applies to a wide range of examples which arise in applications, in particular
in mathematical finance. Some applications are given in Chapters 4
and 5.
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 (Chapter 3).
This forward equation generalizes the Dupire equation, originally derived in
the case of diffusion models, to the case of a discontinuous semimartingale.
1.2.1 Chapter 2 : Markovian projection of semimartingales
Consider, onafilteredprobabilityspace(Ω,F,(Ft)t≥0,P), anItosemimartingale,
on the time interval [0,T], T > 0, given by the decomposition
ξt = ξ0+
t
0
βsds+
t
0
δsdWs+
t
0
y≤1
y ˜ M(dsdy)+
t
0
y>1
yM(dsdy),
where ξ0 is in R d , W is a standard R n -valued Wiener process, M is an
integer-valued random measure on [0,T]×R d with compensator measure µ
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