Special models: Corollary 3.1.7 Examples
Proof of Lemma 3.2.1: By the Stone-Weierstrass theorem,
C
2
K is dense in
C
K for each finite
⊂ . Pick a bijection between and the positive integers and fix a point z
∈ K . Define, for x ∈ K ,
π
n
x : = x
1
, x
2
, . . . , x
n
, z, z, . . ..
3.2.1 The sets πK
are uniformly dense in K , and since each f
∈
C
K is uni-
formly continuous, it is the uniform limit of functions f
n
x : = f π
n
x 3.2.2
depending on finitely many coordinates. Hence
C
fin
K is dense in
C
K .
To see that A satisfies the maximum principle, fix f ∈
C
2 fin
K and suppose
that f assumes its maximum in a point x. Fix an i ∈ . Keeping all x
j j
=i
fixed, f assumes its maximum as a function of the remaining variable in the point x
i
. By the convexity of K it is easily checked that
j α
a j − ix
α j
− x
α i
∂ ∂
x
α i
f x ≤ 0.
3.2.3 Condition 3.1.8 ensures that
αβ
w
αβ
x
i ∂
2
∂ x
α i
∂ x
β i
f x ≤ 0,
3.2.4 as can be seen by writing the matrix wx in diagonal form:
αβ
w
αβ
x
i ∂
2
∂ x
α i
∂ x
β i
f x =
˜α
λ
˜α
x
∂
2
∂ x
˜α 2
f x 3.2.5
for an appropriate orthonormal basis x
˜α
of
R
d
. By condition 3.1.8, the only non-zero terms in 3.2.5 occur for directions that lie in the space I
x
, and for such directions the second derivative is non-positive.
We equip K with the Borel σ -field generated by the open sets. We write
D
K
[0, ∞ for the cadlag functions from [0, ∞ to K
, equipped with the metric d from chapter 3, section 5 of [16], which generates the Skorohod topology. By
C
K
[0, ∞ we denote the continuous functions from [0, ∞ to K
. On
D
K
[0, ∞
we choose the Borel σ -field generated by the open sets of this topology. We equip the space of probability measures on
D
K
[0, ∞ with the topology of weak con-
vergence and we denote weak convergence of the laws of processes with sample paths in
D
K
[0, ∞ by ⇒. Thus X
n
⇒ X means that E[ f X
n
] → E[ f X]
as n → ∞
3.2.6 for all bounded continuous real functions f on
D
K
[0, ∞. By a solution to the
martingale problem we always mean a solution with sample paths in
D
K
[0, ∞.
Lemma 3.2.2 For each probability measure on K there exists a solution to the
martingale problem for A with initial condition µ. Each solution to the martingale problem for A has sample paths in
C
K
[0, ∞. The space of solutions to the mar-
tingale problem for A is compact in the topology of weak convergence. If X
n
, X
solve the martingale problem for A, then X
n
⇒ X implies X
n
t ⇒ X t for all
t ≥ 0.
Proof of Lemma 3.2.2: Existence of solutions to the martingale problem for A follows from Lemma 3.2.1 in combination with Theorem 5.4 and Remark 5.5 from
chapter 4 of [16].
The continuity of sample paths can be shown by Problem 19 from the same chapter: for this one needs to find for every x
∈ K a function f
x
∈
D
A such that for every ε 0
inf { f
x
y − f
x
x : x, y ∈ K
, dx, y
≥ ε} 0 3.2.7
and such that lim
x →y
A f
x
y = A f
y
y = 0 for all x ∈ K
. Instead of working with A, one may also use the closure of A. Applying Lemma 3.4.5 below and
defining γ
i i
∈
as in 3.2.14, it is not hard to check that the functions f
x
y : =
i
γ
i
|x
i
− y
i
|
3
3.2.8 satisfy the requirements.
Compactness of the space of solutions follows from Lemma 5.1 and Re- mark 5.2 from chapter 4 of [16]. Finally, weak convergence in path space of solu-
tions X
n
to the martingale problem for A implies convergence of finite-dimensional distributions by Theorem 7.8 from chapter 3 of [16] and the continuity of sample
paths.
Proof of Theorem 3.1.1: Existence of solutions to the martingale problem for A is guaranteed by Lemma 3.2.2. Corollary 3.4 from chapter 5 of [16] generalizes in a
straightforward way to the infinite-dimensional case, and so for each solution to the martingale problem for A we can find a weak solution to the stochastic differential
equation 3.1.2.
We next show that for each shift-invariant initial condition µ, there exists a shift-invariant solution to 3.1.2. It suffices to construct a shift-invariant solution
to the martingale problem for A. We define a shift operation on
D
K
[0, ∞ in the
obvious way, by putting
T
j
x
i
t : = x
i − j
t i, j
∈ , t ≥ 0. 3.2.9
Let X be a solution to the martingale problem for A with initial condition
L
X 0 = µ. By Lemma 3.3.3 below, there exists a sequence of functions
p
n
: → [0, ∞ such that
i
p
n
i = 1 for each n and
lim
n →∞
k
| p
n
i − k − p
n
j − k| = 0
∀i, j ∈ . 3.2.10
Let X
n
be a sequence of processes with sample paths in
D
K
[0, ∞ with law
L
X
n
=
k
p
n
k
L T
k
X . 3.2.11
Then each X
n
solves the martingale problem for A with initial condition
k
p
n
kµ ◦ T
−1 k
= µ, where we use that µ is shift-invariant. By Lemma 3.2.2 we can find a subsequence X
nm
and a solution X
∞
to the martingale problem for A such that X
nm
⇒ X
∞
. Clearly X
∞
has initial condition
L
X
∞
= µ and for any bounded continuous real function f on
D
K
[0, ∞ we have
|E[ f
T
j
X
nm
] − E[ f X
nm
] |
=
k
p
nm
kE[ f
T
j
T
k
X ] −
i
p
nm
kE[ f
T
k
X ] =
k
p
nm
k − jE[ f
T
k
X ] −
i
p
nm
kE[ f
T
k
X ] ≤
k
| p
nm
k − j − p
nm
k | f
∞
. 3.2.12
By 3.2.10 it follows that
T
j
X
∞
and X
∞
have the same distribution as a proba- bility measure on
D
K
[0, ∞, which implies that their finite-dimensional distribu-
tions agree. Hence X
∞
is shift-invariant.