Environmental Engineering Reference
In-Depth Information
ξ
sn
(
t
) causes a pulse in d
In fact, in Eq. (
2.57
), each pulse in
φ/
d
t
and hence a jump
in
φ
. This fact has the important consequence that the value of
φ
that needs to be
used in
g
(
φ
) is undetermined.
Ito's convention
assigns the rule that
g
(
φ
) should be
calculated with the value of
just before the jump (
van Kampen
,
1981
). This is a
very simple convention that, however, has the consequence that the standard rules of
calculus do not apply anymore to stochastic equations with multiplicative WSN such
as Eq. (
2.57
). For example, if we define a transformed variable
z
φ
=
φ
φ
,the
resulting stochastic equation under Ito's convention is (see
van Kampen
,
1981
)
φ
)d
/
1
g
(
g
[
d
z
d
t
=
f
[
φ
(
z
)]
1
2
φ
(
z
)]
(
z
)]
+
ξ
sn
(
t
)
(
z
)]
−
,
(2.58)
g
[
φ
g
2
[
φ
rather than
d
z
d
t
=
f
[
φ
(
z
)]
+
ξ
sn
(
t
)
,
(2.59)
g
[
φ
(
z
)]
as we would expect from the ordinary rules of calculus.
For a physicist, this mathematical peculiarity of Ito's convention may cause some
interpretation problems: Consider an equation similar to (
2.57
) but with a noise term
with short, though not null, autocorrelation time
c
. For example, we can consider the
case of DMN (see Section
2.2
). In this case no interpretation rule would be required
for Eq. (
2.57
) (see Subsection
2.2.3
and
van Kampen
,
1981
) and the standard rules
of calculus would apply, leading to Eq. (
2.59
) in the limit for
τ
0. The physicist's
interpretation (
van den Broeck
,
1983
) of the white noise as the limit of a colored noise
with
τ
→
c
τ
→
0 would therefore be undermined by use of Ito's convention. The rule that
directly arises from the use of the ordinary rules of calculus is called the
Stratonovich
rule (e.g.,
van Kampen
,
1981
). In this topic we adopt in most cases the Stratonovich
interpretation of Eq. (
2.57
) because, as noted, it is physically more plausible and it
allows us to establish a more direct relationship between processes forced by white
and colored noises. Examples of the implications of adopting Ito's interpretation are
also given for some specific cases in the following chapters.
The Stratonovich rule is often defined as the rule that calculates
g
(
c
φ
)byusinga
value of
before and after the jump
(e.g.
van Kampen
,
1981
;
van den Broeck
,
1997
). However, this definition is valid
only when the driving force is a white Gaussian noise (see Section
2.4
), whereas
it would lead to some inconsistent results in the case of WSN. This is not just a
matter of definition: The knowledge of the correct size of the jump in the
φ
obtained as half of the sum of the values of
φ
trajectory
is essential to numerically simulate a process driven by WSN. We therefore need
to clarify this incongruence. If we want the ordinary rules of calculus to apply to
Eq. (
2.57
), we have from (
2.57
) that the relation between the value
φ
φ
0
before the jump
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