Continuing the discussion of I1, I2, I3, I4, it appears that the output of this definition of stochastic integral is a random entity ; perhaps a process which is some collection of random variables .
Again comparing this with basic integration of a real number‐valued function , the integral is some kind of average or weighted aggregate value for . This integral, if it exists, produces a single unique real number (depending on the value of t), denoted by .
For random variable‐valued integrand , suppose (for the purpose of speculation) that the stochastic integral
(if it exists) is equivalent (in some unspecified sense) to a single, unique random variable . Remember, a random variable is a function, usually real‐valued5, defined on a sample space . Two such functions, and , are the same function if and only if
Does the definition of stochastic integral in I1, I2, I3 yield such a unique value for ? I2 and I3 do not guarantee uniqueness: there may be different sequences in I2 which converge “in mean square” to . In effect, I4 asserts weak convergence of the integrals of the step functions to a value for the integral of , that value being not necessarily unique.
If the integral does not have a unique value, what connections may exist between alternative values? Suppose there is more than one candidate random variable, say and , for the value of the stochastic integral,
In that case, what is the relation between and ? For instance, is it the case that, for each real number a, the probabilities of corresponding measurable sets are equal (such as ):
The framework outlined above does not include the important case , where () is Brownian motion. Broadly speaking, means that the random variables represented by finite sums
converge as tend to zero, each j. In fact the convergence is weak, not point‐wise, with
