Projection markovienne de processus stochastiques

tel-00766235, version 1 - 17 Dec 2012
[40, Theorem 7.1, Chapter 4] implies that for any x0 ∈ Rd , if (X,Qt0,x0)
isasolutionofthemartingaleproblemforL,thenthelawofηt = (t,Xt)
under Qt0,x0 is a solution of the martingale problem for A: in particular
for any f ∈ C∞ 0 (Rd ), γ ∈ C1 ([0,∞) ) and ∀t ≥ t0,
t
qt0,t(x0,dy)f(y)γ(t) = f(x0)γ(0)+ qt0,s(x0,dy)A(fγ)(s,y)ds.
(2.11)
[40, Theorem 7.1, Chapter 4] implies also that if the law of ηt = (t,Xt)
is a solution of the martingale problem for A then the law of X is also
a solution of the martingale problem for L, namely: uniqueness holds
for the martingale problem associated to the operator L on C ∞ 0 (R d ) if
and only if uniqueness holds for the martingale problem associated to
the martingale problem for A on D 0 . Denote C0([0,∞[×R d ) the set of
continuous functions on [0,∞)×R d and vanishing at infinity. Define,
for t ≥ 0 and h ∈ C0([0,∞[×R d ),
∀(s,x) ∈ [0,∞[×R d , Uth(s,x) = Qs,s+t(h(t+s,.))(x). (2.12)
The properties of Qs,t then imply that (Ut,t ≥ 0) is a family of linear
operators on C0([0,∞[×R d ) satisfying UtUr = Ut+r on C0([0,∞[×R d )
and Uth → h as t ↓ 0 on D 0 . (Ut,t ≥ 0) is thus a contraction semigroup
on C0([0,∞[×R d ) satisfying a continuity property on D 0 :
t0
∀h ∈ D 0 , lim
t↓ǫ Uth(s,s) = Uǫh(s,x). (2.13)
3. We apply [40, Theorem 2.2, Chapter 4] to prove that (Ut,t ≥ 0) is
a strongly continuous contraction on C0([0,∞[×R d ) with infinitesimal
generator given by the closure A of A. First, observe that D 0 is dense
in C0([0,∞[×R d ). The well-posedness of the martingale problem for
A implies that A satisfies the maximum principle. It is thus sufficient
to prove that Im(λ − A) is dense in C0([0,∞[×R d ) for some λ where
Im(λ−A) denotes the image of D0 under (λ−A).
4. Without loss of generality, let us put t0 = 0 in the sequel.
For h ∈ D 0 , the martingale property yields
∀0 ≤ ǫ ≤ t < T, ∀(s,x) ∈ [0,T]×R d ,
Uth(s,x)−Uǫh(s,x) =
32
t
ǫ
UuAh(s,x)du.
(2.14)