Feller Processes and Semigroups

6 Feller Processes and Semigroups
Q for every h ∈ H. This can imply that for a.e. ω, X t (ω) has right limit along Q. If not, you can choose a, b
to be the different limits of different sequences from the right, then choose h ∈ H which separates a, b, thus
we can get two different limits h(a) and h(b) of two different sequences for h(X t ), which is a contradiction
with the existence of right limit for h(X t ).
Now, set ˜X t (ω) := lim r↓t,r∈Q X r (ω) for those ω the right limit exists; ˜Xt (ω) ≡ 0 for those ω the right limit
doesn’t exist.
I claim that ∀ t, ˜Xt = X t a.s. To prove this, take any bounded g, h ∈ Ĉ,
E π (g(X t )h( ˜X t )) =
lim E π(g(X t )h(X s ))
s↓t,s∈Q
= lim
s↓t,s∈Q E π(g(X t )T s−t h(X t )) (condition on F t )
= E pi (g(X t )h(X t )).
So, for any positive Borel function f(x, y) on Ŝ × Ŝ, we have E πf(X t , ˜X t ) = E π f(X t , X t ). By the property
of Ŝ, 1 x≠y is such a function, so ˜X t = X t a.s. for any t.
Now, ∀ h ∈ H, use the lemma again, we have for a.e. ω, h( ˜X t ) is right continuous and has left limits along
Q for every h ∈ H because e −λt R λ f( ˜X t ) is again right continuous supermartingale. This implies for a.e. ω,
the ˜X t (ω) is right continuous and has the left limit along Q. By some soft calculus arguments, ˜Xt (ω) must
be cadlag for a.e. ω.
To see those two remainders, notice when X t or X t− = ∆, those supermartingales must be 0 after then,
which means X t has to be ∆ after then, so is ˜X t . If T t , t ≥ 0 are all conservative, π(S) = 1, then X t ∈ S for
any t, so is ˜X t , and all the regularity doesn’t change.
Now, we can take Ω to be space of all Ŝ valued cadlag function s.t. ∆ is absorbing. Under any P π, (X) t is
a Markov process with initial π, transition kernels µ t , and has cadlag path. Particularly, X ≡ ∆ on [ζ, ∞],
where
ζ := inf{t ≥ 0 : X t = ∆orX t− = ∆}
Take (F t ) to be right continuous filtration induced by (X t ), shift operators θ t .
Definition 27.12 (canonical Feller process). Process (X t , t ≥ 0) with distribution P π , filtration (F t ),
shift operators θ t , and cadlag path is called the canonical Feller process with semigroup (T t , t ≥ 0).
To show the strong Markov property, we use the similar arguments as for Brownian motion.
Theorem 27.13 (strong Markov property). For any canonical Feller process (X t , t ≥ 0) with initial π,
stopping time τ, and r.v. Y ,
E π (Y ◦ θ τ |F τ ) = E Xτ Y a.s.-P π on {τ < ∞}
Proof. Let (G t ) be the filtration induced by (X t ). Let
τ n := [2n τ] + 1
2 n
then, all τ n are G-stopping time and F τ ⊂ G τn
have strong Markov property for each τ n , i.e.
for any n. Notice τ n only takes countably many value, so we
E π (Y ◦ θ τn ; A) = E π (E Xτn ; A)
∀ A ∈ F τ , n ∈ N