Let $X$ be a cadlag $L^{p}$ martingale ($p>1$). Let $q$ be the Hölder conjugate of $p$. Let $F$ be a finite subset of $[0,t]$. The following claim appears in a proof of Doob's maximal inequality that I am reading: $$ \mathbb{E}\left(\max_{s\in F}\left|X_{s}\right|^{p}\right)\leq q^{p}\max_{s\in F}\mathbb{E}\left(\left|X_{s}\right|^{p}\right)\leq q^{p}\mathbb{E}\left(\left|X_{t}\right|^{p}\right). $$ The first inequality comes from the discrete version of Doob's maximal inequality, but I don't understand where the last inequality comes from. Hints?
Asked
Active
Viewed 882 times
2
-
3I think it is from that $|X|^p$ is sub martingale (by Jensen's inequality). – Jay.H May 06 '16 at 13:44
-
2You are correct: $\mathbb{E}[|X_{t}|^{p}]=\mathbb{E}[\mathbb{E}[|X_{t}|^{p}\mid\mathcal{F}{s}]] \geq \mathbb{E}[\mathbb{E}[|X{t}|\mid\mathcal{F}{s}]^p]=\mathbb{E}[|X{s}|^{p}]$ thanks :-) – user303911 May 06 '16 at 13:51
1 Answers
0
As suggested by Jay.H, the solution follows from Jensen's inequality: $$ \mathbb{E}\left[\left|X_{t}\right|^{p}\right]=\mathbb{E}\left[\mathbb{E}\left[\left|X_{t}\right|^{p}\mid\mathcal{F}_{s}\right]\right]\geq\mathbb{E}\left[\mathbb{E}\left[\left|X_{t}\right|\mid\mathcal{F}_{s}\right]^{p}\right]=\mathbb{E}\left[\left|X_{s}\right|^{p}\right]. $$
user303911
- 53