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

Hoover, Douglas N.; Perkins, Edwin

doi:10.1090/S0002-9947-1983-0678335-1

Nonstandard construction of the stochastic integral and applications to stochastic differential equations. II
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by Douglas N. Hoover and Edwin Perkins

Trans. Amer. Math. Soc. 275 (1983), 37-58

DOI: https://doi.org/10.1090/S0002-9947-1983-0678335-1

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Part I: Trans. Amer. Math. Soc. (1) (1983), 1-36

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 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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