Consider a random walk on $\mathbb{Z}$ with triangular drift and jumps that are standard normals. That is, $$ \begin{cases} RW_{t+1} = RW_t - d + \epsilon_t, \quad t \geq 0,\\ RW_{t-1} = RW_t - d + \epsilon_{t-1}, \quad t \leq 0, \end{cases} $$ where $\epsilon_t$ are iid standard normals.

Using Skorohod embedding, we can think of $W_t$ as the values of a Brownian motion with triangular drift at integer time points: $$ RW_t = B_t - d |t|, \quad \text{for $t \in \mathbb{Z}$} $$ where $B_t$ is a standard Brownian motion (and equality is in distribution). This tells me that $$ \max_{t \in \mathbb{Z}} RW_t \leq \max_{t \in \mathbb{R}} (B_t - d|t|) $$ (in the sense of stochastic domination), and the difference between them is not too large (~max of a Brownian bridge with different endpoints?).

Question: is something similar true for $\arg\max$, the location of the maximum? That is, is it true that $$ \arg\max_{t \in \mathbb{Z}} RW_t \quad \text{ is not too far from } \quad \arg\max_{t \in \mathbb{R}} (B_t - d|t|) ? $$ (Stochastic domination seems unlikely.)

Note that this might have nothing to do with probability -- it might be a question about how continuous the $\arg\max$ functional is.

Consider a random walk on $\mathbb{Z}$ with triangular drift and jumps that are standard normals. That is, $$ \begin{cases} RW_{t+1} = RW_t - d + \epsilon_t, \quad t \geq 0,\\ RW_{t-1} = RW_t - d + \epsilon_{t-1}, \quad t \leq 0, \end{cases} $$ where $\epsilon_t$ are iid standard normals.

Using Skorohod embedding, we can think of $W_t$ as the values of a Brownian motion with triangular drift at integer time points: $$ RW_t = B_t - d |t|, \quad \text{for $t \in \mathbb{Z}$} $$ where $B_t$ is a standard Brownian motion (and equality is in distribution). This tells me that $$ \max_{t \in \mathbb{Z}} RW_t \leq \max_{t \in \mathbb{R}} (B_t - d|t|) $$ (in the sense of stochastic domination), and the difference between them is not too large (~max of a Brownian bridge with different endpoints?).

Question: is something similar true for $\arg\max$, the location of the maximum? That is, is it true that $$ \arg\max_{t \in \mathbb{Z}} RW_t \quad \text{ is not too far from } \quad \arg\max_{t \in \mathbb{R}} (B_t - d|t|) ? $$ (Stochastic domination seems unlikely.)

Update: as Martin and Ofer point out, on a fixed sample path the maximizers can be arbitrarily far from each other. Nonetheless, is there a true statement of the form $$ \arg\max_{t \in \mathbb{Z}} RW_t \leq 2 \cdot \arg\max_{t \in \mathbb{R}} (B_t - d|t|) + 3, $$ for some values of 2 and 3? (In the sense of stochastic domination, of course.)