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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An extremal property of stochastic integrals

Author: Steven Rosencrans
Journal: Proc. Amer. Math. Soc. 28 (1971), 223-228
MSC: Primary 60.75
MathSciNet review: 0275535
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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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Keywords: Nonanticipating Brownian functional, stochastic integral, Itô’s lemma, Bessel process, moments, parabolic equation, hyperbolic equation
Article copyright: © Copyright 1971 American Mathematical Society