5.3 Choosing a variation h
ξ t
h
ξ t
h
ξ t
As discussed in [HM06] and at length in [Mat08], if one looks for the variation h
ξ
such that 44 holds and
R
t
|h
ξ s
|
2
ds is minimized, then the answer is h
ξ s
= A
∗ t
M
−1 t
J
t
ξs which by the observation in 31 is simply h
ξ s
= G
∗
K
s,t
M
−1 t
J
t
ξ. While this is not quite the correct optimisation problem to solve since its solution h
ξ
is not adapted to W and hence E
| R
t
h
ξ s
· dW s|
2
6= R
t
E |u
s
|
2
ds, it is in general a good enough choice.
A bigger problem is that the space on which M
t
can be inverted is far from evident. If the range of G was dense in
H which requires infinitely many driving Wiener processes, then there is some chance that RangeJ
t
⊂ RangeM
t
and the above formula for h
t
could be used. This is in fact the case where the Bismut-Elworthy-Li formula is often used and which might be refereed to as “truly elliptic.”
It this case the system is in fact strong Feller. We are precisely interested in the case when only a finite number of directions are forced or the variance decays so fast that this is effectively true. One
of the fundamental ideas used in this article is that we need only effective control of the system on a finite dimensional subspace since the dynamic pathwise control embodied in Assumption B.4 can
control the remaining degrees of freedom.
While Theorem 6.7 of the next section gives conditions that ensure that M
t
is almost surely non- degenerate, it does not give much insight into the structure of the range since it only deals with finite
dimensional projections. However, Assumption B.1 ensures that it is unlikely the eigenvectors with sizable projection in Π
H have small eigenvalues. As long as this is true, the “regularised inverse” M
t
+ β
−1
, which always exists since M
t
is positive definite, will be a “good inverse” for M
t
, at least on Π
H . This suggests that we make the choice h
ξ s
= G
∗
K
s,t
M
t
+ β
−1
J
t
ξ for some very small β 0. Observe that
D
ξ
u
t
− D
h
ξ
u
t
= J
t
ξ − M
t
M
t
+ β
−1
J
t
ξ = βM
t
+ β
−1
J
t
ξ , 50
which will be expected to be small as long as J
t
ξ has small projection relative to the size of β in Π
⊥
H . But in any case, the norm of the right hand side in 50 will never exceed the norm of J
t
ξ, so that for small values of
β, kD
ξ
u
t
− D
h
ξ
u
t
k is expected to behave like kΠ
⊥
J
t
ξk. Assumption B.4 precisely states that if one projects the Jacobian onto Π
⊥
H , then the system behaves as if it was “strongly dissipative” as in Section 5.1.1. All together, this motivates alternating between
choosing h
ξ
= A
∗ n,n+1
M
n,n+1
+ β
n −1
J
n,n+1
ρ
n
for even n and h
ξ
≡ 0 on [n, n + 1] for odd n. Since we will split time into intervals of length one, we introduce the following notations:
J
n
= J
n,n+1
, A
n
= A
n,n+1
, M
n
= M
n,n+1
. We then define the map
ξ, W 7→ h
ξ
W recursively by h
ξ s
= A
∗ 2n
β
2n
+ M
2n −1
J
t
ρ
2n
s for s
∈ [2n, 2n + 1 and n ∈ N ,
for s ∈ [2n − 1, 2n and n ∈ N .
51 Here, as before,
ρ = ξ, ρ
t
= J
0,t
ξ − A
0,t
h
ξ s
= D
ξ
u
t
− D
h
ξ
u
t
, and β
n
is a sequence of positive random numbers measurable with respect to
F
n
which will be chosen later. Observe that these definitions are not circular since the construction of h
ξ s
for s ∈ [n, n + 1 only
requires the knowledge of ρ
n
, which in turn depends only on h
ξ s
for s ∈ [0, n. The remainder of this
690
section is devoted to showing that this particular choice of h
ξ
is “good” in the sense that it allows to satisfy 49. We are going to assume throughout this section that Assumptions A.1 and B.1-B.4 hold,
so that we are in the setting of Theorem 5.5, and that h
ξ
is defined as in 51.
5.4 Preliminary bounds and definitions
Kategori