# Extending the Burkholder-Davis-Gundy for continuous local martingales using localization

Proposition 3.26 below is from Karatzas and Shreve's Brownian Motion and Stochastic Calculus, which gives a preliminary result of the Burkholder-Davis-Gundy Inequality. The proposition assumes that $$M$$ is a continuous martingale with $$M$$ and $$\langle M \rangle$$ bounded. The B-D-G inequality gives (3.26) for $$m>0$$ with only the requirement that $$M$$ be a continuous local martingale.

Question: Remark 3.27 states that a straightforward localization argument shows that (3.27) and (3.29) are valid for any continuous local martingale $$M$$. Indeed, we could consider the stopping time $$T_n= \inf \{t\ge 0: |M_t| + \langle M \rangle_t \ge n\}$$, which tends to $$\infty$$ and gives that $$(M^{T_n}_t)_{t\ge 0}$$ is a bounded martingale. I can see that then by taking $$n \to \infty$$ and using monotone convergence, we would get (3.27) and (3.29) for $$M \in \mathscr{M}^{c,loc}$$, however, I don't understand why we wouldn't get (3.28) without the additional condition $$E(\langle M \rangle_T^m)<\infty$$.

Why do we require this condition? In fact, as we can see from the last bit of the proof of 3.26, we get (3.29) from (3.27) and (3.28), so I can't figure out why we wouldn't just get (3.28) for continuous local martinagles without this additional condition.

Example: $$M$$ is standard Brownian motion started at $$0$$, and $$T$$ is the first hitting time of state 1, and $$m=1$$. Then the RHS of (3.28) is $$1$$ while the LHS is $$+\infty$$.
If $$(T_n)$$ is a localizing sequence of bounded stopping times (so that $$M$$ stopped at $$T_n$$ is bounded by $$n$$, say) then (3.28) gives $$B_1E(T_n)\le E(|M_{T_n}|^2),$$
for each $$n$$. The LHS tends to $$+\infty$$ by monotone convergence, hence so must the RHS.