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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.

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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.

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