I'm attempting to read a book on stochastic calculus by D.H. Fremlin, which is the 6th volume of his treatise on measure theory encompassing all kinds of topics related it.
Before I make a very significant time investment, I'd be curious about how much his way to setup stochastic integration differs from more traditional and standard approaches, in two ways:
- Is his approach to construct it non-mainstream?
- is the final outcome perhaps a slightly different mathematical object then the standard one (e.g. in Sondermann D. - Introduction to Stochastic Calculus for Finance. A New Didactic Approach it seems to me that a different non-mainstream construction is used, but which nevertheless leads to the standard outcome)? Resp. how does it compare with the theory as it is setup in Shalizi C. - Almost None of the Theory of Stochastic Processes?)
It seems to me that the author bases his construction on measure algebras, which he introduces in volume 3. My level of measure theory is somewhere along his volume 2. While I do not mind reading up things from volume 3 to follow his exposition on stochastic calculus, I would very much mind if I found that afterwards I can't talk with some of my colleagues anymore, who to a course in stochastic calculus that went the traditional way, because we each speak "language" that is not quite the same, since the text I read was too non-standard. Is there such a danger?
Here is an excerpt from the first chapter of volume 6 (called chapter 61 there): I begin with an attempt to give a coherent and complete description of the principal form of stochastic integration which will be investigated in this volume. As elsewhere in probability theory, it is customary to set this material out in terms of ordinary random variables, that is, measurable functions defined on probability spaces. We find immediately, however, that while integrands and integrators may well present themselves most naturally in this form, the integrals we construct are defined, in the cases for which this theory has been developed, in terms of convergence in $|| \ ||_1$ or $|| \ ||_2$ or in measure, and therefore correspond not to definable functions, but to equivalence classes of functions. Moreover, integrands and integrators can be changed on negligible sets without affecting the values of the corresponding integrals. I believe that the theory becomes clearer and cleaner if we move directly to operations on evolving families in $L^0$ . While this demands an initial investment by the reader in a more abstract framework for the ideas of elementary probability theory, the translation is not difficult, and a full exposition can be found in Chapter 36
Here is another excerpt from chapter 4 (i.e. chapter 64):
[We will develop] what I will call the ‘S-integral’ (645P). In fact the S-integral is much closer than the Riemann-sum integral to the standard stochastic integral developed in [Protter 05].
