Showing papers on "Semimartingale published in 1983"

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

Abstract: H. J. Keisler has recently used a nonstandard theory of lto integration (due to R. M. Anderson) to construct solutions of Ito 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, c) = h(t, c) + f(s, , y(, w)) dz(s), where z is a given semimartingale, h is a right-continuous process and f(s, w, ) 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. Introduction. This paper is a continuation of Hoover and Perkins [5]. Notation and definitions presented in [5] are used without further introduction, and references to that paper are made by simply citing the number of the theorem or lemma (due to the consecutive numbering of the two papers, no ambiguity will arise). Finally, the reader is referred to ?0 of [5] for a brief description of the contents of this work. 8. Continuous local martingales. We assume throughout this section that all processes are R or *R-valued, since the main results (Theorems 8.1, 8.5 and 8.6) then follow immediately for higher dimensions. We will show (Theorem 8.1) that a continuous local martingale has a 'q1-local martingale lifting for any internal filtration {( t} (recall that, in general, Theorem 5.6 asserted only that such a lifting exists for some internal filtration). For this reason, unless otherwise stated, the internal filtration under consideration will be {(it I t E T} (which was used in the original definition of {T}). Note that if X: T X Q -*M is an S-continuous process (i.e., X(*, w) is S-continuous a.s.) then st( X) has continuous paths a.s. and X is a uniform lifting of st(X) in the sense of Keisler [6]. The following lifting theorem was first proved in a more specialized setting in Panetta [9]. The proof given here is different. Received by the editors January 22, 1982. The results of this paper were presented at the second Victoria Symposium on Nonstandard Analysis, Victoria, Canada, June 16-20, 1980. A MS (MOS) subject cllassifications (1970). Primary 60H20, 02H25; Secondary 60G45.

Stochastic equations and krylov's estimates for semimartingales

Abstract: In 1969 N. V. Krylov obtained the following estimate where X is an Ito process in is the exit time of X from a bounded region is Ld-norm of a measurable nonnegative function , and N is a constant. We generalize estimates of this type to semimartingales and give applications to the theory of stochastic equations with respect to semimartingales. The questions of the existence, uniqueness, convergence and comparison of solutions of these equations are also studied.

Semimartingales gaussiennes ― application au problème de l'innovation

Abstract: Recently N.C. Jain and D. Monrad [8] have obtained a decomposition of the paths of a gaussian quasimartingale into a martingale and a predictable process of integrable variation such that these components are jointly gaussian. In the first part of this paper we prove the same result for a gaussian semimartingale. In the second part we give some applications to the innovation problem.

Interchanging the order of stochastic integration and ordinary differentiation

Abstract: We give conditions under which the order of differentiation w.r.t. a Euclidian parameter and stochastic integration w.r.t. a continuous semimartingale can be interchanged.

On weak convergence of semimartingales to stochastically continuous processes with independent and conditionally independent increments

Abstract: The authors study weak convergence of a sequence of semimartingales to an arbitrary stochastically continuous process independent or conditionally independent increments. The "semimartingale scheme" they consider includes the traditional "series scheme". Bibliography: 22 titles.

Fonctions convexes et semimartingales

Abstract: If ƒ is the difference of two convex functions on R n and X is a semimartingale, then ƒ(X) is also a semimartingale. We study the converse in this paper.