I have been posting my stochastic calculus notes on this blog for some time, and they have now reached a reasonable level of sophistication. The basics of stochastic integration with respect to local martingales and general semimartingales have been introduced from a rigorous mathematical standpoint, and important results such as Ito’s lemma, the Ito isometry, preservation of the local martingale property, and existence of solutions to stochastic differential equations have been covered.
I will now start to also post examples demonstrating results from stochastic calculus, as well as counterexamples showing how the methods can break down when the required conditions are not quite met. As well as knowing precise mathematical statements and understanding how to prove them, I generally feel that it can be just as important to understand the limits of the results and how they can break down. Knowing good counterexamples can help with this. In stochastic calculus, especially, many statements have quite subtle conditions which, if dropped, invalidate the whole result. In particular, measurability and integrability conditions are often required in subtle ways. Knowing some counterexamples can help to understand these issues. The following points demonstrate some of the issues which I plan to address with the upcoming examples.
- It is often assumed that we work with respect to a filtered probability space which is complete. This was required for results such as the debut theorem and the existence of cadlag versions of martingales and of stochastic integrals. If the completeness condition is dropped then none of these results hold, and it would be useful to have some examples demonstrating what can go wrong.
- Local martingales naturally arise in stochastic calculus as integrals of martingales. It is easy to construct examples which are not proper martingales. However, many easily constructed examples are either quite artificial or it is clear that the martingale property breaks down. On the other hand, the martingale property does also fail in many natural cases which are likely to be encountered in simple and practical stochastic models, but where it is not obvious that this is happening. I shall give some examples demonstrating this.
- We know that an integral of a bounded elementary process with respect to a martingale is itself a martingale. More generally, for locally bounded integrands, stochastic integration preserves the local martingale property. It is natural to ask whether, for bounded integrands, it also preserves the martingale property. This statement is known to hold in many cases, such as for Lp-integrable martingales (any
) and, more generally, for any martingale whose maximum process is integrable. This covers the cases which are likely to be encountered most of the time. However, the statement does sometimes fail, and I will post some examples where this happens.
- In order to define a stochastic integral satisfying some basic properties, it is necessary for the processes we work with to be semimartingales. Not only do such processes include all (adapted) finite variation processes and all martingales, there are some very general conditions ensuring that a process is a semimartingale. However, there are also some natural processes for which these conditions fail, and which are not semimartingales. The standard techniques of stochastic calculus will, therefore, not apply to such processes.
Almost Sure 
Dear George,
now that you have introduced the stochastic integral and SDEs, I was wondering if you were planning to describe some models where these ideas naturally show up? Finance, stochastic control, statistical physics, diffusion approximation, etc … I am always impressed by the quality of your posts and would be very interested in knowing you thoughts on that!
Best,
Alekk
Hi. As you bring it up, I will at some point try describing some models implementing the theory. Probably in finance, but possibly other stuff too. For the time being I’ve still got a lot more ideas of maths I want to post than I have time to actually spend writing it. So, I’m not planning on posting on finance models very soon. But I’ll definitely give it some thought.
btw, I see you’ve set up a blog. Good stuff – I added it to my blogroll.
Hi George,
Can you solve this one? Given a stochastic function F(X(t),t) where dX(t) is an increment of a Brownian motion, if X(0)=0 evaluate the integral (from 0 to t) of t^2sinXdX(t)
I’ve been having some problems with it so any help would be most appreciated — thanks!
H
Sounds like a rather arbitrary question. Why are you interested in this? Is it an exercise?
Anyway, I’d consider trying a substitution Y = cos(X) and writing it in terms of dY (using Ito’s lemma), then use integration by parts to write it as a function of Y(t) and an integral over dt. If the integral over dt doesn’t drop out then I don’t think there is an answer of the form F(X(t),t).
Dear John, how can I differentiate this process?
$X_{t}=\int_{0}^{t}\exp{\delta (B_{t}-\mu t -B_{s}+\mu s)-\frac{1}{2}(t-s)ds}$
Where $B_{t}$ is a standart Brownian motion and $\mu, \delta$ are real numbers
Can you help me?
Hi George,
Maybe a counterexample to add to this collection (unless if it’s trivial) would be the answer (yet to be found) to the question asked on MSE here : https://math.stackexchange.com/questions/3458652/c1-functions-of-continuous-semimartingale Regards
I posted an answer to the question, thanks for pointing it out. It is probably worth a short blog post.