Nonstandard construction of the stochastic integral and applications to stochastic differential equations. II
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 by Douglas N. Hoover and Edwin Perkins PDF
 Trans. Amer. Math. Soc. 275 (1983), 3758 Request permission
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 \[ y(t,\omega ) = h(t,\omega ) + \int _0^t {f(s,\omega ,y(\cdot ,\omega )) dz(s)} \], where $z$ is a given semimartingale, $h$ is a rightcontinuous process and $f(s,\omega , \cdot )$ is continuous on the space of rightcontinuous 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.References

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Additional Information
 © Copyright 1983 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 275 (1983), 3758
 MSC: Primary 60H10; Secondary 03H05
 DOI: https://doi.org/10.1090/S00029947198306783351
 MathSciNet review: 678335