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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Nonstandard construction of the stochastic integral and applications to stochastic differential equations. II


Authors: Douglas N. Hoover and Edwin Perkins
Journal: Trans. Amer. Math. Soc. 275 (1983), 37-58
MSC: Primary 60H10; Secondary 03H05
Part I: Trans. Amer. Math. Soc. (1) (1983), 1-36
MathSciNet review: 678335
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Abstract: H. J. Keisler has recently used a nonstandard theory of Itô integration (due to R. M. Anderson) to construct solutions of Itô integral equations by solving an associated internal difference equation. In this paper we use the same general approach to find solutions $ y(t)$ of semimartingale integral equations of the form

$\displaystyle y(t,\omega) = h(t,\omega) + \int_0^t {f(s,\omega ,y(\cdot ,\omega))\,dz(s)} $

, where $ z$ is a given semimartingale, $ h$ is a right-continuous process and $ f(s,\omega , \cdot)$ is continuous on the space of right-continuous functions with left limits, with the topology of uniform convergence on compacts. In addition, we generalize Keisler's continuity theorem and give necessary and sufficient conditions for an internal martingale to be $ S$-continuous.

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DOI: https://doi.org/10.1090/S0002-9947-1983-0678335-1
Keywords: Stochastic differential equations, semimartingale, nonstandard analysis
Article copyright: © Copyright 1983 American Mathematical Society