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In page 45 of the book "Financial Derivatives In Theory and Practice by P.J.Hunt and J.E.Kennedy, it seems to me that the author says the stopped Brownian Motion is not a martingale as follows.

(Quote)

Does the martingale property

M(t)=E[M(T)|F(t)]

$$M(t)=E[M(T)|F(t)]$$

hold if T $T$ is a stopping time? In general the answer is no, as can be seen by taking M to be Brownian Motion and T=inf{t>0: M(t)>=1} $T=\inf\{t>0: M(t)\ge1\}$ (Unquote)

I do not understand why the martingale property does not hold in this case and appreciate any explanation on this.

1

# Is stopped brownian motion not a martingale ?

In page 45 of the book "Financial Derivatives In Theory and Practice by P.J.Hunt and J.E.Kennedy, it seems to me that the author says the stopped Brownian Motion is not a martingale as follows.

(Quote)

Does the martingale property

M(t)=E[M(T)|F(t)]

hold if T is a stopping time? In general the answer is no, as can be seen by taking M to be Brownian Motion and T=inf{t>0: M(t)>=1} (Unquote)

I do not understand why the martingale property does not hold in this case and appreciate any explanation on this.