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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An extremal property of stochastic integrals
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by Steven Rosencrans PDF
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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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 28 (1971), 223-228
  • MSC: Primary 60.75
  • DOI: https://doi.org/10.1090/S0002-9939-1971-0275535-7
  • MathSciNet review: 0275535