This is a well know conclusion of the boundary condition of lookback option. Here $$\dfrac{d S_t}{S_t} = (\mu - D)dt + \sigma dW_t$$ is underlying asset. $$M_t^{(n)} =\left[\int^t_0S^n_u d u\right]^{\dfrac{1}{n}}$$ $$M_t = \max\limits_{t_0\leq u \leq t}S_u$$ is maximal process of $S_t,$ and we have the conclusion: $$\lim\limits_{n\rightarrow\infty}M_t^{(n)} = M_t$$ $V(S_t,t,M_t)$ is the price of lookback option at time $t.$

One of the boundary condition of PDE for $V$ is $$\dfrac{\partial V}{\partial M_t}\big|_{S_t = M_t} = 0$$ this is to make the $d M_t$ term zero.

But, some books explain this condition as following way, do you think it is reasonable? enter image description here


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