Weak Solutions for Semi-Martingales

Canadian journal of mathematics, Tome 33 (1981) no. 5, pp. 1165-1181

Résumé

The fundamental theorem of this paper is stated in Section 8. In this theorem, the stochastic differential equation dX = a(X)dZ is studied when Z is a *-dominated (cf. [15]) Banach space valued process and a is a predictable functional which is continuous for the uniform norm.For such an equation, the existence of a “weak solution” is stated; actually, the notion of weak solution here considered is more precise than this one introduced by Strook and Varadhan (cf. [30], [31], [23]).Namely, this weak solution is a probability, so-called “rule,” defined on (D H × Ω), D H being the classical Skorohod space of all the cadlag sample paths and Ω is the initial space which Z is defined on: the marginal distribution of R on Ω is the given probability P on Ω. This concept of rule is defined in Section 3.

Détail

DOI : 10.4153/CJM-1981-088-9

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