  Famously mathematical finance was started by Bachelier in his 1900 PhD thesis
where - among many other achievements - he also provides a formal derivation of
the Kolmogorov forward equation. This forms also the basis for Dupire's (again
formal) solution to the problem of finding an arbitrage free model calibrated
to the volatility surface. The later result has rigorous counterparts in the
theorems of Kellerer and Lowther. In this survey article we revisit these
hallmarks of stochastic finance, highlighting the role played by some optimal
transport results in this context.
