Before reading this I must not I think I am a little behind on some of the prereq for this topic but I really want to be able to understand it in a relatively meaningful way.
I am having trouble getting through the idea to a stochastic integral and could really use some help in understanding it. I would like to write out a good summary(perhaps not too heavy on proofs) for the notes for my course. So far this is my thinking:
My understanding is than we begin with idea idea that the total variation of Brownian paths are infinite a.s. so thus we cannot define them in the Riemann-Stiietjes sense.
I would then like to go on to explain that we can integrate for for simple functions where
\begin{equation} f(t) = \sum_{j=0}^{N-1}a_jI_{(t_{j},t_{j+1}]}(t). \end{equation}
with $a_j$ non-random and $I$ the indicate function. By taking limits we could then integrate a wider class of functions but they would still not be random. From there I would go on to say we can integrate for functions where $a_j$ is random but I do not understand why we can do this. This is where i assume we would differentiate between Ito and Strat due to the the point $t^*_k$ that we choose, is this correct place?
Finally by taking by taking a sequence of functions of the above type we can define the integral for a "wider" (what does this mean)? class of functions. I am also finding it hard to understand the sense in which they converge.
I am having some trouble getting these ideas into my head and I really hope someone can give me some more details that will help me, I have read through a variety of books but still end up quite confused. I would like to understand this in quite a lot more detail but am becoming increasingly confused and frustrated with my results. I would also like to end by describing the properties of these integrals.
Convergence is both $L^2$ and in probability, and if you take a class of subdivisions small fast enough, you even have convergence a.s.
When introducing randomness, do not forget to introduce measurable random processes notions.
The construction is just an extension of an isometry from a dense subspace to the whole space, here for $L^2$ functions.