Projection markovienne de processus stochastiques

tel-00766235, version 1 - 17 Dec 2012
defined, for f ∈ C ∞ 0 (Rd ), by
Ltf(x) = b(t,x).∇f(x)+
+
R d
d
i,j=1
aij(t,x)
2
∂2f (x)
∂xi∂xj
[f(x+y)−f(x)−1{y≤1}y.∇f(x)]n(t,dy,x),
(2.3)
where a : [0,T]×R d ↦→ Md×d(R), b : [0,T]×R d ↦→ R d and
n : [0,T]×R d ↦→ R(R d −{0}) are measurable functions.
For (t0,x0) ∈ [0,T]×R d , we recall that a probability measure Qt0,x0 on
(Ω0,BT) is a solution to the martingale problem for (L,C ∞ 0 (Rd )) on [0,T] if
Q(Xu = x0, 0 ≤ u ≤ t0) = 1 and for any f ∈ C ∞ 0 (R d ), the process
f(Xt)−f(x0)−
t
t0
Lsf(Xs)ds
isa(Qt0,x0,(Bt))-martingaleon[0,T]. Existence, uniqueness andregularityof
solutions to martingale problems for integro-differential operators have been
studied under various conditions on the coefficients [90, 59, 40, 66, 77, 42].
We make the following assumptions on the coefficients:
Assumption 2.1 (Boundedness of coefficients).
(i) ∃K1 > 0, ∀(t,z) ∈ [0,T]×R d
1∧y 2
, b(t,z)+a(t,z)+
n(t,dy,z) ≤ K1
(ii) lim
R→∞
T
sup
0 z∈Rd n(t,{y ≥ R},z) dt = 0.
where . denotes the Euclidean norm.
Assumption 2.2 (Continuity).
(i) For t ∈ [0,T] and B ∈ B(R d − {0}), b(t,.), a(t,.) and n(t,B,.) are
continuous on R d , uniformly in t ∈ [0,T].
(ii) For all z ∈ R d , b(.,z), a(.,z) and n(.,B,z) are continuous on [0,T[,
uniformly in z ∈ R d .
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