alt="t Subscript j minus 1 Baseline less-than-or-equal-to s less-than t Subscript j"/>, . (Accordingly, in I1, can be regarded as a “degenerate” random variable, with atomic probability value.) Suppose the integrator is the real‐valued ds instead of the random variable‐valued . Then4
Formally, at least, this looks like the definition in I1 of when is a step function. The factor equals for each j. This emerges naturally from the mathematical meaning of the length or distance variable s, and from the mathematical meaning of .
Can this be replicated in I1 when is a step function, or when each is a fixed real number ? Is it the case that
With each , this would imply
(1.1)
If this is unproblematical, it should be possible to deduce it from one or other of the various mathematical definitions of Brownian motion , along with some mathematical definition of the integral in this context.
But it appears that there is no such understanding of . So, as in I1, it seems that this formulation is to be regarded as a basic postulate or axiom of stochastic integration.
Returning to the definition of the classical Itô integral, I2 has the following condition on the expected value of the integral of the process :
The idea here is that, if is the random entity obtained by carrying out some form of weighted aggregation—denoted by —of all the individual random variables (), then
This formulation assumes that the aggregative operation , involving infinitely many random variables (), produces a single random entity whose expected value can be obtained by means of the operation .
Additionally, is said to be a Lebesgue integral‐type construction. The Скачать книгу
alt="t Subscript j minus 1 Baseline less-than-or-equal-to s less-than t Subscript j"/>, 