It's a good observation.

Actually you will not hesitate to apply Ito formula to C(S,t) since you know C is driven by dW(t). While actually the maximizing and average process are the same. If you take a close look of the definition, you will find that:

M(t) (maximizing) A(t) (average) process are F(t) measurable not F(T) (e.g. M(t)=max(0<u<t)W(u)). Which means that M(t) is the max number up to now (t), not on the whole time period(T). By this definition, it ensures that the process is driven by dW(t) just as the stock or option price (i.e. S(t) and C(S,t)) which means they are Ito process since they can be written into ...+∫b(s)dW(s) so you can safely apply Ito formula to them.

best,

In fact M(t) is not Ito process.
1.If it is ,dM(t) = A(t)dt + B(t)dWt
2.according to (7.4.11) , B(t) = 0;
3.accoding to (7.4.12), A(t) does not exist.

the author is misusing Ito, he just wants to have a PDE

Maybe you are not right

a yes, I made a mistake, m(t) is a continuous function.

But I think the existence of dy does not need it to be a Ito prccess. Ito formula is nothing but an application of Taylor expansion. You are applying Ito to v, but y is not necessary an Ito.  For example, it can be a possion jump, in the last chapter of the book, you will see the Ito formula with possion jump.
And I think Ito formula can be applied in many situations. Not just on Ito process.

I am not sure whether I catch your point. But it's a really good observation, I have not thought about it before.

Best,

作者在书中一直坚持用“微分”的观点解释随机分析，直观但不严谨，容易误导没学过随机分析的人（特别是学微积分但不学泛函的人），“微分”只不过是一种简写形式，实质应该是“积分”，关于“鞅”的积分是一个特殊的“线性算子”，一种随机积分的构造就是基于线性算子理论。随机分析中研究“半鞅”的积分，Ito公式只是一个“有漂亮结果”的特例。


如果阁下有机会，请你帮我联系作者，把我的问题发送给他，看看作者怎么说，谢谢