Fix a time horizon $T>0$. Consider a fixed discretization parameter $\Delta>0$ and divide $[0,T]$ into intervals of the form $[K \Delta, (K+1)\Delta]$ for $K=0...\lfloor T/ \Delta \rfloor - 1$. Now consider a family of continuous stochastic processes on $[0,T]$, $\{Z^\epsilon\}_{\epsilon>0}$, depending on a small parameter $\epsilon>0$. In particular assume that $Z_t^\epsilon$ satisfies some SDE on $[0,T]$ with coefficients depending on $\epsilon$. Assume that $\{Z^\epsilon\}_{\epsilon>0}$ is continuous but its definition is different in each $[K \Delta, (K+1)\Delta)$
Now assume I can prove that that there exists $C>0$ (independent of $\epsilon, K$) such that $$\mathbb{E}\left [\sup_{t \in [K \Delta, (K+1)\Delta ]} |Z_t^\epsilon| \right ]\leq C \epsilon $$
In this context I was wondering if I can prove that there exists $\bar C>0$ (independent of $\epsilon, K$) such that $$\mathbb{E}\left [\sup_{t \in [0,T ]} |Z_t^\epsilon| \right ]\leq \bar C \epsilon $$ Of course we have $$\mathbb{E}\left [\sup_{t \in [0,T ]} |Z_t^\epsilon| \right ]=\mathbb{E}\left [\max_{K=0...}\sup_{t \in [K \Delta, (K+1)\Delta ]} |Z_t^\epsilon| \right ] \geq \mathbb{E}\left [\sup_{t \in [K \Delta, (K+1)\Delta ]} |Z_t^\epsilon| \right ] $$ so we cannot immediately use our hypothesis but can I somehow get my thesis?
Set $M = \lfloor T / \Delta \rfloor - 1$. Consider the following inequality: $$\sup_{t \in [0,T]} |Z_t^\epsilon| = \max_{K = 0, \ldots , M} \sup_{t \in [K \Delta, (K+1)\Delta]} |Z_t^\epsilon| \leq \sum_{K=0}^M \sup_{t \in [K \Delta, (K+1)\Delta]} |Z_t^\epsilon|$$ which is, morally, simply $\|\cdot\|_\infty \leq \|\cdot\|_1$ in $\mathbb{R}^M$. Taking expectations, $$\mathbb{E}\left( \sup_{t \in [0,T]} |Z_t^\epsilon| \right) \leq \sum_{K=0}^M \mathbb{E}\left( \sup_{t \in [K \Delta, (K+1)\Delta]} |Z_t^\epsilon| \right) \leq (M+1)C \epsilon$$ Which shows that setting $\bar{C}=(M+1)C$ suffices to get the desired bound.