Using Ito theory to decide whether $M^f$ is martingale or a local martingale

Question

I came across the following while reading Ikeda & Watanabe book Stochastic differential equations and Diffusion processes, in page 163-164

At first the sentence $$f(X_t)- f(X_0) - \int_0^t Af(s,X)\, ds \in \mathcal{M}^{c,loc}_2 $$ seems strange, since $f \in C^2_b$ implies that $f, \partial_i f, \partial_i \partial_j f $ are bounded functions, so in principle the local martingale must be bounded (for each $t$) wich gives us that it is a true martingale and we should have $$f(X_t)- f(X_0) - \int_0^t Af(s,X)\, ds \in \mathcal{M}^{c}_2 $$

But then, on second thought, we don't know if the terms $\alpha^i_k(s,X)$ are bounded so we must recognize that the first sentence was true.

Now when we move to the other expression, we read that (after correcting some typos) $$M^{(l)}_i(t) = X^i(t\wedge \sigma_l) - X^i(0) - \int_0^{t \wedge \sigma_l} \beta^i(s,X)\, ds \in \mathcal{M}^c_2$$

But the only thing we know is that $X_{s \wedge \sigma_l}$ is on a bounded set, the functions $\alpha^i_k(s,X), \beta^i(s,X)$ might still be unbounded.

So it seems that $$M^{(l)}_i(t) = X^i(t\wedge \sigma_l) - X^i(0) - \int_0^{t \wedge \sigma_l} \beta^i(s,X)\, ds \in \mathcal{M}^{c,loc}_2$$

What is going wrong?

Answer 1

Concerning the unboundedness of $\alpha$ and $\beta$: you are absolutely right, there is some inaccuracy in Ikeda-Watanabe. In order for the argument to work, one needs to impose a stronger assumption e.g. the one from Remark 1.1, Chapter 4:

For any $T>0$ and $M>0$ $$ \sup_{t\in[0,T],\|w\|_T\le M} \big(\|\alpha(t,w)+ \|\beta(t,w)\|\big) <\infty. $$

Then for $t\le \sigma_l$, $\alpha(t,X)$ and $\beta(t,X)$ will be bounded, so the process $M$ stopped at $\sigma_l$ is a martingale, as you correctly note (all processes will be bounded). Therefore, $M$ is a local martingale with the localizing sequence $\{\sigma_l,l\ge 1\}$.