# From maximal inequality in finite time to continuous time

In class we have seen a maximal inequality about discrete sub martingale. The setting was the following : Consider $$T=\{0,1,..,N\}$$ and $$(X_t)_{t\in T}$$ a sub martingale. Then we have for all $$\lambda>0$$

$$\lambda\mathbb{P}(\{\max_{t\in T}X_t\geq\lambda\}\leq\mathbb{E}(X_{T}1_{\max_{t\in T}X_t\geq\lambda})$$

I would like to extend it to continuous martingale. Here is my attempt.

First, after a first attempt I decided to consider a martingale because at some point, for convergence issue (you will tell me if I am wrong) I will need to have an increasing sequence of functions to apply monotone convergence. This leads to the use of absolute value and unfortunately this does not preserve a sub martingale.

Now let $$T\in(0,\infty)$$. Define the set $$D_n = \{\frac{kT}{2^n} : k = 0,…,2^n\}$$, clearly it is an increasing sequence of sets that equals $$[0,T]$$ at the limit.

Next define $$Y_n(\omega) = \sup_{d\in D_n}\lvert X_d(\omega)\rvert$$, it is an increasing sequence which converges almost surely to $$Y(\omega) = \sup_{t\in [0,T]}\lvert X_t(\omega)\rvert$$. This implies that

$$1_{Y_n\geq\lambda}\to_{n\to\infty}1_{Y\geq\lambda}$$

Applying the maximal inequality for finite time set to $$\lvert X_d\rvert$$ we get, for all $$n$$

$$\lambda\mathbb{P}(\{Y_n\geq\lambda\}\leq\mathbb{E}(\lvert X_{T}\rvert1_{Y_n\geq\lambda})$$

Then by convergence monotone and the continuity of the probability (which can be used here since $$X$$ is continuous in $$t$$) we have

$$\lambda\mathbb{P}(\{Y\geq\lambda\}\leq\mathbb{E}(\lvert X_{T}\rvert1_{Y\geq\lambda})$$

I would like to know if it is correct please and if not what can be done to improve this attempt.

Thank you a lot !

Your approach seems okay, so long as the submartingale $$X$$ has paths $$t\mapsto X_t(\omega)$$ that are right continuous with left limits. Then for each sample point $$\omega$$, $$\sup_{t\in[0,T]}X_t(\omega) =\lim_n\sup_{t\in D_n}X_t(\omega),$$ the limit on the right being monotone increasing. The absolute values aren't necessary.