Consider the probability space $(\Omega, \mathcal{F},\mathbb{P})$, and let $X:\Omega \mapsto \mathbb{R}^T$, where $T$ is an index set, be a random function. What is the canonical definition of the optima of $X$?
The simplest I could think of is $$\mathrm{optima}(X) := \mathrm{optima}\left(t \mapsto \mathbb{E}[X(t)]\right).$$ However, this definition ignores all the randomness in $X$. I'd instead want the definition to satisfy something like the following
If $t_1$ and $t_2$ are two candidate optima such that $\mathbb{E}[X(t_1)] = \mathbb{E}[X(t_2)]$ and $\mathbb{Var}[X(t_1)] < \mathbb{Var}[X(t_2)]$, then $t_1$ is a better choice than $t_2$.
and
If $t_1$ and $t_2$ are two candidate optima such that $\mathbb{Var}[X(t_1)] = \mathbb{Var}[X(t_2)]$ and $\mathbb{E}[X(t_1)] > \mathbb{E}[X(t_2)]$, then $t_1$ is a better choice than $t_2$.
which is to say that I prefer points with higher certainty. To this end, I could for example define $$\mathrm{optima}(X) := \mathrm{optima}\left(t \mapsto \frac{\mathbb{E}[X(t)]}{\sqrt{\mathbb{Var}[X(t)]}} \right),$$ or $$\mathrm{optima}(X) := \mathrm{optima}\left(t \mapsto \mathbb{E}[X(t)] - \sqrt{\mathbb{Var}[X(t)]} \right).$$
But these seem contrived. I suspect there are other better/standard definitions in common use, that I just haven't come across.
For a concrete example, one may use $$X_t = t(1-t) + B_t, \quad t \in [0,1]$$ where $B_t$ is standard Brownian motion, and define its maxima. By the first definition above, the maxima is $\dfrac{1}{2}$, where as by the second it is $\dfrac{1}{3}$.
Some references I've found: