An extremal property of stochastic integrals

In this paper we consider the stochastic integral y t = ∫ 0 t e ( s , b ) d b s {y_t} = \int _0^t {e(s,b)d{b_s}} of a nonanticipating Brownian functional e e that is essentially bounded with respect to both time and the Brownian paths. Let f f be a convex function satisfying a certain mild growth condition. Then E f ( y t ) ≦ E f ( | | e | | b t ) Ef({y_t}) \leqq Ef(||e||{b_t}) , where b t {b_t} is the position at time t t of the Brownian path b b . As a corollary, sharp bounds are obtained on the moments of y t {y_t} . The key point in the proof is the use of a transformation, derived from Itô’s lemma, that converts a hyperbolic partial differential equation into a parabolic one.