5'4 Stratonovich Stochastic Integral

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5.4 Stratonovich Stochastic Integral The second
definition of a stochastic integral
Stratonovich. Why necessary? Ito integral nice
mathematical properties, e.g. connection Ito
SDE's lt-gt diffusion processes. Ito integral
implies that there is no dependence between the
process Xt and the random force Wt at the same
instant of time t. Choosing the left points of
the partition as evaluation points exploits,
maximally the fact that the Wiener process has
independent increments. While this definition
is certainly the only reasonable one, as already
mentioned, if we deal with truly white noise
one should not forget that for many real
applications white noise is only an idealization.
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Stratonovich integral suited for modeling such
physical systems A SDE with a white-noise term
is obtained, via a limiting procedure. The
starting point is a real noise, with short but
nonzero correlation time. Short but finite
memory of the noise gt dependence between the
process X t and the random force at the
same instant of time t. The limit from the real
noise to the white-noise idealization has to be
effected carefully, such that, roughly speaking,
the impact of the external noise is conserved.
We expect that a trace of the dependence between
X t and should survive in the
white noise limit. Stratonovich integral takes
into account just such a correlation between the
system and the external noise. So to speak it is
the white-noise limit of a real noise problem.
Clearly, this is a problem of the modeling
procedure what SDE and thus what random
process X t describes most appropriately the
system in the white-noise limit (In biological
applications, e. g., insects with non overlapping
generations, a discrete time description is most
appropriate. This is genuine discrete while
noise.)
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5.4.1 Definition of the Stratonovich Integral
Relation with the Ito Integral Give a
precise definition of the Stratonovich integral
So far we have considered only scalar random
processes. However, it will become apparent in
the following that in order to be able to define
-a Stratonovich SDE for a scalar process, we need
to consider the integral definition for a
two-dimensional vector process. This is due to
the fact that the approximating sums of the
Stratonovich integral are slightly anticipating.
In order to find the approximation to
for in the
approximating sum,
it is not sufficient to know the Wiener
process up to time s. Knowledge of the future
behavior, up to is needed, since
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An n-dimensional diffusion process
is defined in a straightforward extension of the
one-dimensional case as a Markov process which
fulfills the conditions
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Let be a two-dimensional diffusion process,
with drift vector
and the diffusion matrix
Let H(x) be a two-dimensional real valued vector
function with continuous partial derivatives
and
and
Then the limit
is called a stochastic integral in the sense of
Stratonovich. It is connected with the Ito
integral, here defined as
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STRATONOVICH ITO
ITO TERM INTEGRAL
INTEGRAL
Proof
5.67

½ ? ds ?xk Htrj Bkj
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The 2 integrals are not equal! Which one holds in
nature for a system is a physical
input. Alternatively, to get the same
quantitative consequences for the physical
system, different parameters for the Ito and
Stratonovich processes have to be used.
In order to define the Stratonovich stochastic
differential equation, we set
In the following, the Stratonovich integral will
always be distinguished from an Ito integral by
writing between integrand and dW We say of
a process X t fulfilling
that it obeys the Stratonovich stochastic
differential equation (S SDE)
5.70
The Ito SDE for the composite diffusion process
reads of course
5.71
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or in matrix notation
where Vt is a dummy Wiener process.
For the diffusion matrix we have
Hence the transformation formula (5.67) yields in
this case
5.74
This implies that the Ito SDE
5.75
is equivalent to the the Stratonovich SDE (5.70)
in the sense that they have one and the same
solution.
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• Let us discuss the main consequences of this
  transformation formula.
• It is always possible to change from the
  Stratonovich interpretation of a SDE to the Ito
  interpretation by adding
• ii) The Stratonovich SDE (5.70) defines (cf 5.75)
  
• a diffusion process with drift
  and diffusion g2.
• iii) In the case of additive noise, i. e., g(x)
  const,
• there is no difference between the Ito and
  the Stratonovich integral.
• iv) In the case of multiplicative noise, i. e.,
  g(x) const,
• gt the influence of the random force depends on
  the state of the process,
• gt implicit correlation between the process Xt
  and the random force Wt,
• leads to a systematic contribution in the
  evolution of Xt.
• It gives rise to the noise induced drift
  .

This term is also known as spurious drift
since it does not
appear in the phenomenological equation. There
is, however, nothing spurious in this term, it
has physical consequences. v) The Stratonovich
calculus obeys the classical chain rule. This
point is verified, using (5.75)
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5.4.2 Ito or Stratonovich A Guide for the
Perplexed Modeler
Importance of the Stratonovich integral models
directly the correlations between the random
environment and the system. These correlations
in the white-noise idealization are a trace of
the stochastic dependence between the state of
the system and the environmental fluctuations
when the latter have a nonvanishing correlation
time. Indeed, X t is a function of the history
of the environment,
is not equal to zero for all t gt 0
So if
then X t and are dependent
stochastic variables.
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Stratonovich's choice models most directly the
physical situation the only definition of a
stochastic integral leading to a calculus with
classical rules Wong-Zakai Theorem. Let
be a sequence of random processes, which
are continuous, of bounded variation, having
piecewise continuous derivative and converging
almost surely uniformly to the Wiener process W
t. Then under some mild conditions on f and g,
the solutions of where all integrals can be
understood as ordinary Riemann integrals,
converge almost surely uniformly to the solution
of the Ito SDE According to the
transformation rule (5.75), the Stratonovich SDE
equivalent to (5.77) is
5.77
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This implies that if we start out with a
phenomenological equation, containing real noise
of the above form, i.e. (5.79) and we
pass to the white-noise limit to obtain an SDE of
the form (5.80) the latter has then
to be interpreted as a Stratonovich equation.
For practical calculations, it is convenient to
switch over from the S SDE to Ito (5.77). The
main advantage of the Ito version (5.77) of SDE
(5.80) the direct display of the
characteristics of the diffusion process X t
that models a system coupled to an environment
with extremely rapid fluctuations the drift
consists of the phenomenological part f(x) plus
the noise-induced s2 g' g/2 the latter
indicates that in a fluctuating environment the
systematic motion is modified too if the noise is
only approximately white. The diffusion is
given by that part of the phenomenological
equation that multiplies the fluctuating
parameter.
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The unfamiliar situation that the calculus of
differential equations with white-noise
coefficients can be based on different
definitions of the stochastic integrals, involved
in solving such equations, has led to an
astonishingly long lived controversy in the
physical literature as to which is the right
definition. A lot of confusion has been
created. We have the impression that quite a few
authors were led astray by at least implicitly
assuming some special intrinsic value for the
classical rules of calculus. They have, however,
none. Often a principle of in variance of the
equations under "coordinate transformation" y
u(x) is invoked to pick the Stratonovich integral
as the "right" one and reject the Ito integral as
the "wrong" one. It should be realized that
this principle refers to an invariance of the
form of the SDE under a nonlinear transformation
of the system. Though it is a particular feature
of the S SDE to be invariant, there is no
physical virtue in this form invariance. It is
only a different guise of the fact that the
Stratonovich calculus obeys the familiar
classical rules. The only quantities that have to
be invariant under a transformation y u(x),
where u is one to one, are the probabilities, as
for instance
This is of course guaranteed in both calculi.
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The above presentation of the Ito and
Stratonovich integrals shows clearly that there
is no question which of the two integrals is the
right one both lead to a consistent calculus.
The question is, how to interpret an SDE with
Gaussian white noise that has been obtained via
some limiting procedure, or to rephrase the
question, what are the coefficients of the
diffusion process that models most adequately the
system we want to describe. It is at this point
that physical arguments have to be invoked to
determine the appropriate drift and diffusion
coefficient of the process. Formulated in this
way, it becomes clear that the question Ito
versus Stratonovich is void, as also stressed by
van Kampen 5.10 a diffusion process can be
described either way. The Ito versus Stratonovich
controversy has also been dealt with in a clear
way by Jazwinski 5.11, p. 140 "One might argue
that the Stratonovich integral should be used
because it is simpler, since it can be
manipulated by the formal rules. This is
intuitively appealing but beside the point. The
Stratonovich integral does not offer any new
mathematical insight or content. As a matter of
fact, all the results concerning the Stratonovich
integral are proved using Ito's theory. Of
itself, the Stratonovich integral does not offer
any additional physical insight. It does, under
certain conditions, more directly model a
physical process" and "Most important of all is
the fact that the Ito integral is defined for a
much broader class of functions than the
Stratonovich integral."
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Clearly the basic problem is not what is the
right integral definition, but how do we model
real systems by diffusion processes. If the
starting point is a phenomenological equation in
which some fluctuating parameters are
approximated by Gaussian white noise, then the
most appropriate diffusion process is the one
that is defined by the Stratonovich
interpretation of this equation. On the other
hand, for a broad class of biological and other
systems, e. g., insect populations with
nonoverlapping generations, the appropriate
starting point is a discrete time description,
such as
(5.82)
and
where
are Gaussian independent random variables with
(5.83) , , and (5.84) If times
are large compared to the generation length
then it is permissible to pass to the
continuous time limit (5.85)
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It is obvious from the asymmetric form of (5.82)
with respect to time that in this case the most
appropriate diffusion process is the one that is
defined by the Ito interpretation of (5.85). 2
rough criteria for the diffusion model for a real
system If the stochastic differential
equation has been obtained as the white-noise
limit of a real noise equation, choose the
Stratonovich interpretation If it represents
the continuous time limit of a discrete time
problem, interpret it according to Ito. In
any case, the ultimate test is the confrontation
of the analytical results with the experimental
facts. There are no universally valid
theoretical a priori reasons why one or the
other interpretation of an SDE should be
preferred.