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 part of this statement should be unproblematical. The domain is a real interval, and has a distance or length function, which, in the context of Lebesgue integration on the domain, gives rise to Lebesgue measure on the space of Lebesgue measurable subsets of . So can also be expressed as .
However, the random variable‐valued integrand is less familiar in Lebesgue integration. Suppose, instead, that the integrand is a real‐number‐valued function . Then the Lebesgue integral , or , is defined if the integrand function f is Lebesgue measurable. So if J is an interval of real numbers in the range of f, the set is a member of the class of measurable sets; giving
