Introduction
Suppose we wish to model a process described by a differential equation and initial condition
But we wish to do this in the presence of noise. It’s not clear how do to this but maybe we can model the process discretely, add noise and somehow take limits.
Let be a partition of then we can discretise the above, allow the state to be random and add in some noise which we model as samples of Brownian motion at the selected times multiplied by so that we can vary the amount noise depending on the state. We change the notation from to to indicate that the variable is now random over some probability space.
We can suppress explicit mention of and use subscripts to avoid clutter.
We can make this depend continuously on time specifying that
and then telescoping to obtain
In the limit, the second term on the right looks like an ordinary integral with respect to time albeit the integrand is stochastic but what are we to make of the the third term? We know that Brownian motion is nowhere differentiable so it would seem the task is impossible. However, let us see what progress we can make with socalled simple proceses.
Simple Processes
Let
where is measurable. We call such a process simple. We can then define
So if we can produce a sequence of simple processes, that converge in some norm to then we can define
Of course we need to put some conditions of the particular class of stochastic processes for which this is possible and check that the limit exists and is unique.
We consider the , the space of square integrable functions with respect to the product measure where is Lesbegue measure on and is some given probability measure. We further restrict ourselves to progressively measurable functions. More explicitly, we consider the latter class of stochastic processes such that
Less Simple Processes
Bounded, Almost Surely Continuous and Progressively Adapted
Let be a bounded, almost surely continuous and progressively measurable process which is (almost surely) for for some positive constant . Define
These processes are cleary progressively measurable and by bounded convergence ( is bounded by hypothesis and is uniformly bounded by the same bound).
Bounded and Progressively Measurable
Let be a bounded and progressively measurable process which is (almost surely) for for some positive constant . Define
Then is bounded, continuous and progressively measurable and it is well known that as . Again by bounded convergence
Progressively Measurable
Firstly, let be a progressively measurable process which is (almost surely) for for some positive constant . Define . Then is bounded and by dominated convergence
Finally let be a progressively measurable process. Define
Clearly
The Itô Isometry
Let be a simple process such that
then
Now suppose that is a Cauchy sequence of progressively measurable simple functions in then since the difference of two simple processes is again a simple process we can apply the Itô Isometry to deduce that
In other words, is also Cauchy in and since this is complete, we can conclude that
exists (in ). Uniqueness follows using the triangle inequality and the Itô isometry.
Notes

We defer proving the definition also makes sense almost surely to another blog post.

This approach seems fairly standard see for example Handel (2007) and Mörters et al. (2010).

Rogers and Williams (2000) takes a more general approach.

Protter (2004) takes a different approach by defining stochastic processes which are good integrators, a more abstract motivation than the one we give here.

The requirement of progressive measurability can be relaxed.
Bibliography
Handel, Ramon von. 2007. “Stochastic Calculus, Filtering, and Stochastic Control (Lecture Notes).”
Mörters, P, Y Peres, O Schramm, and W Werner. 2010. Brownian motion. Cambridge Series on Statistical and Probabilistic Mathematics. Cambridge University Press. http://books.google.co.uk/books?id=eTbAdSrzYC.
Protter, P.E. 2004. Stochastic Integration and Differential Equations: Version 2.1. Applications of Mathematics. Springer. http://books.google.co.uk/books?id=mJkFuqwr5xgC.
Rogers, L.C.G., and D. Williams. 2000. Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus. Cambridge Mathematical Library. Cambridge University Press. https://books.google.co.uk/books?id=bDQyzoHWfcC.