  Consider a directed analogue of the random graph process on $n$ vertices,
where the $n(n-1)$ edges are ordered uniformly at random and revealed one at a
time.
  It is known that w.h.p.\@ the first digraph in this process with both
in-degree and out-degree $\geq q$ has a $[q]$-edge-coloring with a Hamilton
cycle in each color. We show that this coloring can be constructed online,
where each edge must be irrevocably colored as soon as it appears. In a similar
fashion, for the \emph{undirected} random graph process, we present an online
$[n]$-edge-coloring algorithm which yields w.h.p.\@ $q$ disjoint rainbow
Hamilton cycles in the first graph of the process that contains $q$ disjoint
Hamilton cycles.
