Projection markovienne de processus stochastiques

tel-00766235, version 1 - 17 Dec 2012
(see [61, Ch.II,Sec.1]), implying that for any predictable process φt and for
any T > 0:
T
E
0 Rd T
φtM(dtdy) = E
0 Rd
φtµ(dtdy) . (1.8)
Ito semimartingales form a natural class of stochastic processes, sufficiently
large for most applications and possessing various analytical properties
which allow in particular the use of stochastic calculus [81, 61].
An Ito semimartingale may be characterized by its local characteristic
triplet (β,δ,µ), which may be seen as a path-dependent generalization of the
notionofLévy tripletforLévyprocesses. 1 Under someconditionsonthelocal
characteristics of ξ, we will show that the Markovian projection X of ξ may
then be constructed, as in Gyöngy [51], by projecting the local characteristics
of ξ on its state. However, our construction differs from that of Gyöngy:
we construct X as the solution of a martingale problem and ensure that this
construction yields a Markov process X, characterized by its infinitesimal
generator. This regularity of the construction will provide a clear link between
the mimicking process X and the corresponding forward equation and
clarify the link between forward equations and Markovian projections.
1.1.4 Stochastic differential equations and martingale
problems
To construct Markovian mimicking processes for ξ, we will need to construct
solutions to a general ’Markovian-type’ stochastic differential equation with
jumps, given by
∀t ∈ [0,T], Xt = X0 +
+
t
0
t
0
y≤1
b(u,Xu)du+
yÑ(du dy)+
t
0
t
0
Σ(u,Xu)dBu
y>1
yN(du dy),
(1.9)
where (Bt) is a d-dimensional Brownian motion, N is an integer-valued random
measure on [0,T]×R d with compensator n(t,dy,Xt−)dt where
(n(t, .,x),(t,x) ∈ [0,T] × R d ) is a measurable family of positive measures
1 We refer to Jacod & Shiryaev [61, Chapter 4 Section 2] for a complete presentation.
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