An extremal property of stochastic integrals
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- by Steven Rosencrans
- Proc. Amer. Math. Soc. 28 (1971), 223-228
- DOI: https://doi.org/10.1090/S0002-9939-1971-0275535-7
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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.References
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Bibliographic 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