  The online increasing subsequence problem is a stochastic optimisation task
with the objective to maximise the expected length of subsequence chosen from a
random series by means of a nonanticipating decision strategy. We study the
structure of optimal and near-optimal subsequences in a standardised planar
Poisson framework. Following a long-standing suggestion by Bruss and Delbaen
(Stoch. Proc. Appl. 114, 2004), we prove a joint functional limit theorem for
the transversal fluctuations about the diagonal of the running maximum and the
length processes. The limit is identified explicitly with a Gaussian
time-inhomogeneous diffusion. In particular, the running maximum converges to a
Brownian bridge, and the length process has another explicit non-Markovian
limit.
