e saw in the section
(
Gyongy lemma
) the possibility to
replicate one dimensional distributions of a complex process
of the generic
form
with the local volatility process
of much simpler
form
provided that the function
is given by the
relationship
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|
(Gyongy 2)
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The section (
Time series
forecasting
) contains a generic proof that the conditional expectation
minimizes unconditional expectation of the square difference. Speaking in
terms of this section, the function
should be guessed to be a solution of the following minimization
problem:
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|
(Markovian projection)
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where the expectation is unconditional and is taken at the observation time.
We note that the expectation is taken with respect to the original process
and not with respect to
:
we do not evaluate
.
This point is not obvious from considerations of the section
(
Time series forecasting
).
Hence, we proceed with derivation of the
(
Markovian projection
).
We define the
functional
for some stochastic process
and consider the
problem
The solution
has to
satisfy
for any smooth deterministic function
.
We calculate the
derivative:
Next, we use the
operation
consistent with the foundation formulas (
Chain
rule
) and (
Total probability rule
).
We slice the event space by the values of
:
Hence, it suffices to
have
Comparing the last relationship with the (
Gyongy 2
)
we see that indeed, the recipes (
Gyongy 2
) and
(
Markovian projection
) are equivalent.
It is important to note that the class of functions
in the minimization problem (
Markovian
projection
) has to be restricted only by existence requirements for the
process
.
It is under such freedom that the conditional expectation and the minimization
problem are equivalent. For general
the process
is not always analytically tractable. We will be restricting our attention to
some subclasses of functions
that provide analytical tractability. Thus, we will be sacrificing equivalence
to gain analytical tractability.
We present several examples of this technique in the following sections.
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