Patrick Muldowney - Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics

For example, the “distance” between and could be

With such a metric at hand, it may then be possible to define , or , as the limit of the integrals of (integrable) step functions converging to for , as .

Unfortunately, most standard textbooks do not give this point much attention. But for relatively straightforward integrands such as , it should not be too difficult.

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‐valued 5 , 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, I3yield such a unique value for ? I2and I3do not guarantee uniqueness: there may be different sequences in I2which converge “in mean square” to . In effect, I4asserts 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,