imilarly to the
section (
Localization
), we are going to
calculate the interval
.
In the section (
Mean reverting
equation
) we calculated that the
SDE
may be integrated
into
We
continue
The integrals
evaluate
for a standard normal random variable
.
We put the results
together:


(Mean reverting solution)

where we introduced the convenience notations
and
,
,
Based on a precision
,
,
we would like to find two numbers
and
such
that
We
calculate
Let
be a number
s.t.
then
and
The process
is connected to
of the section
(
Solving
one dimensional mean reverting equation
) by the
relationship


(Exponential change)

We aim to set the interval
according
to
However, we still need to make two more modifications:
1. We need to adapt
to the binary structure of our mesh. Similar argument was already made in the
section (
Localization
).
2. We want to make
time independent for smaller difference
.
To achieve the goal 2 we
set
We choose
conservatively wider to preserve precision. Adaptive basis selection is a
compensating procedure that preserves efficiency.
For the goal 1, we choose a scale parameter
and
The following is a consequence of the formulas
(
Mean reverting solution
) and
(
Exponential
change
):


(Analytical solution for mean reverting equation)

