e approximate the solution
of the
SDEs
with the solution
of the
SDEs
The function
is restricted to allow for existence of the solution
.
We scale the deterministic function
so that
for some
.
The
is to be chosen to have the property
and to be optimal in the sense of the following proposition.

Proposition

Define
for
and consider solutions
and
of the following
SDEs
The
functional
has the
properties
The particular function
that minimizes
has the
property

(Parameter averaging)

Proof

We calculate the
directly
Hence,
Observe
that
Therefore,
Note that
(by the form of SDEs for
and
),
hence
The above expression for
should be minimized with respect to
.
Hence, we differentiate with respect to the
and equate to
0:
Finally,
and the claim follows.