An extremal property of stochastic integrals

Rosencrans, Steven

doi:10.1090/S0002-9939-1971-0275535-7

An extremal property of stochastic integrals
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Proc. Amer. Math. Soc. 28 (1971), 223-228 Request permission

Abstract:

In this paper we consider the stochastic integral ${y_t} = \int _0^t {e(s,b)d{b_s}}$ of a nonanticipating Brownian functional $e$ that is essentially bounded with respect to both time and the Brownian paths. Let $f$ be a convex function satisfying a certain mild growth condition. Then $Ef({y_t}) \leqq Ef(||e||{b_t})$, where ${b_t}$ is the position at time $t$ of the Brownian path $b$. As a corollary, sharp bounds are obtained on the moments of ${y_t}$. 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.

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Stochastic characteristics for parabolic equations