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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(\vert\vert e\vert\vert{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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  • [1] H. P. McKean, Jr., Stochastic integrals, Probability and Math. Statist., no. 5, Academic Press, New York, 1969. MR 40 #947. MR 0247684 (40:947)
  • [2] R. Courant, Methods of mathematical physics. Vol. 2: Partial differential equations, Interscience, New York, 1962. MR 25 #4216.
  • [3] K. Itô and H. P. McKean, Jr., Diffusion processes and their sample paths, Die Grundlehren der math. Wissenschaften, Band 125, Academic Press, New York; Springer-Verlag, Berlin, 1965. MR 33 #8031. MR 0199891 (33:8031)
  • [4] M. Zakai, Some moment inequalities for stochastic integrals and for solutions of stochastic differential equations, Israel J. Math. 5 (1967), 170-176. MR 37 #991. MR 0225397 (37:991)
  • [5] R. Hersh, Explicit solution of a class of higher-order abstract Cauchy problems (to appear). MR 0270210 (42:5102)
  • [6] L. R. Bragg and J. W. Dettman, Related problems in partial differential equations, Bull. Amer. Math. Soc. 74 (1968), 375-378. MR 37 #560. MR 0224961 (37:560)
  • [7] N. P. Romanov, On one-parameter groups of linear transformations. I, Ann. of Math. (2) 48 (1947), 216-233. MR 8, 520. MR 0020218 (8:520c)
  • [8] R. Hersh and R. Griego, Random evolutions--theory and applications (to appear).
  • [9] A. V. Balakrishnan, Abstract Cauchy problems of the elliptic type, Bull. Amer. Math. Soc. 64 (1958), 290-291. MR 21 #4294. MR 0105555 (21:4294)
  • [10] E. D. Conway, Stochastic characteristics for parabolic equations, Tulane University (multilithed).

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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

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