I was presented with the following result, and it was stressed that this was an important result:
Let $\{ X_t\}_{t \in \mathbb{N}}$ be a martingale. Then, for any $p>1$ and $n \in \mathbb{N}$, we have that $E\left[\max_{1 \leq j \leq n}\mid X_j \mid ^p \right] \leq \left(\frac{p}{1-p}\right)^p E[|X_n|^p]$.
As I understand this, the statement says that the p-th moment of the maximal value of the martingale up to time n is bounded by the p-th moment of the value at time n.
However, I fail to realize the depth and importance of this result. Specifically, I wonder:
- Is this result important for proving other results in stochastic analysis?
- Are there any intuitive examples where this result is used to gain understanding of a stochastic process?