  It is strange but fruitful to think about the functions as random processes.
Any function can be viewed as a martingale (in many different ways) with
discrete time. But it can be useful to have continuous time too. Processes can
emulate functions, expectation of profit functional on the solution of
stochastic differential equation can emulate the functional on usual familiar
functions. The advantage is that now we have "all" admissible functions
"enumerated" as solutions of one stochastic differential equation, and choosing
the best function optimizing a given functional becomes a problem of choosing
the right control process. But such problem has been long since solved in the
part of mathematics called Stochastic Optimal Control. So-called Bellman
equation reduces an infinite dimensional problem of choosing the best control
to a finite dimensional (but non-linear as a rule) PDE called Bellman equation.
Its solution, called Bellman function of a given optimization problem, gives us
a lot of information about optimum and optimizers. This method gave some
interesting results in the classical Harmonic Analysis, having on the surface
nothing to do with probability. Sometimes the results obtained by this method
did not find "classical" proofs so far. It is especially well-suited to
estimates of singular integrals, probably because of the underlying
probabilistic structure of classical singular integrals.
