Projection markovienne de processus stochastiques

tel-00766235, version 1 - 17 Dec 2012
Forx0 inR d , aprobabilitymeasureQx0 on(Ω0,BT)issaidtobeasolution
to the martingale problem for (L,C ∞ 0 (Rd )) on [0,T] if Q(X0 = x0) = 1 and
for any f ∈ C ∞ 0 (R d ), the process
f(Xt)−f(x0)−
t
0
Lsf(Xs)ds
is a (Qx0,(Bt)t≥0)-martingale on [0,T].
If for any x0 ∈ R d , there exists a unique solution Qx0 to the martingale
problem for (L,C ∞ 0 (Rd )) on [0,T], then we say that the martingale problem
for (L,C ∞ 0 (Rd )) on [0,T] is well-posed.
Ethier&Kurtz[40, Chapter4, Theorem4.2]showthatthewell-posedness
of the martingale problem for (L,D(L) implies the Markov property of X
under Q. Hence when one wants to build a Markovian semimartingale
whose infinitesimal generator is an integro-differential operator such as L,
the existence and uniqueness of solutions to martingale problems for integrodifferential
operators are crucial.
Existence anduniqueness for integro-differential operator have been studied
under various conditions on the coefficients. When L has constant coefficients
b(t,x) = b, a(t,x) = a and n(t,dy,x) = n(dy), then L is the
infinitesimal generator of a pure jump Lévy process (see Bertoin [19], Sato
[84]). The existence of a solution to the martingale problem for the operator
L with continuous or measurable coefficients and nondegenerate diffusion
matrix has been considered by Skorokhod [87], Stroock and Varadhan [90],
Krylov [68], Lepeltier and Marchal [72] and Jacod [59]. The uniqueness for
a continuous nondegenerate diffusion coefficient was studied by Stroock and
Varadhan [90], Komatsu [66] and Stroock [88]. In all these results, boundedness
of coefficients guarantees existence, while uniqueness is based on some
form of continuity of coefficients plus an ellipticity condition of the diffusion
matrix. Figalli [42] extends these results to the case of second-order
differential operators with irregular coefficients.
Komatsu [67] was the first to treat the martingale problem for pure jump
processes generated by operators where a = 0. In the case where b and n
are non-time dependent Komatsu [67] proved the existence and uniqueness
of solutions for Lévy Kernels n(dy,x) which are perturbations of an α-stable
Lévy measure. Uniqueness in the case of a state dependent singularity at
zero was proved by Bass [10, 11] under some uniform continuity the intensity
of small jumps with respect to the jump size.
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