Showing papers on "Semimartingale published in 1995"

On the minimal martingale measure and the möllmer-schweizer decomposition

Abstract: We provide three characterizations of the minimal martingale measure[Pcirc] associated to a given d-dimensional semimartingale X. In each case, [Pcirc] is shown to be the unique solution of an optimization problem where one minimizes a certain functional over a suitable class of signed local martingale measures for X. Furthermore, we extend a result of Ansel and Stricker on the Follmer-Schweizer decomposition to the case where X is continuous, but multidimensional.

Stock Price Returns and the Joseph Effect: A Fractional Version of the Black-Scholes Model

Abstract: In mathematical finance, semimartingales are traditionally viewed as the largest class of stochastic processes which are economically reasonable models for stock price movements. This is mainly because stochastic integrals play a crucial role in the modern theory of finance, and semimartingales represent the largest class of stochastic processes for which a general theory of stochastic integration exists. However, some empirical evidence from actual stock price data suggests stochastic models that are not covered by the semimartingale setting.

Perfect cocycles through stochastic differential equations

Abstract: We prove that if ϕ is a random dynamical system (cocycle) for whicht→ϕ(t, ω)x is a semimartingale, then it is generated by a stochastic differential equation driven by a vector field valued semimartingale with stationary increment (helix), and conversely. This relation is succinctly expressed as “semimartingale cocycle=exp(semimartingale helix)”. To implement it we lift stochastic calculus from the traditional one-sided time ℝ to two-sided timeT=ℝ and make this consistent with ergodic theory. We also prove a general theorem on the perfection of a crude cocycle, thus solving a problem which was open for more than ten years.

Stratonovich stochastic differential equations driven by general semimartingales

Abstract: On considere des equations stochastiques differentielles ou le «bruit» est une semimartingale quelconque (avec des sauts). On propose une interpretation des integrales stochastiques du type «Stratonovich», mais du genre de celles introduites par S. Marcus, plutot que du genre de celles de P. A. Meyer. On etablit l'existence et l'unicite des solutions et on demontre que les flots sont des diffeomorphismes quand les coefficients sont convenables (ce qui n'est pas le cas pour l'interpretation de Meyer-Stratonovich). De plus on etablit les proprietes de Markov fortes, et on demontre un genre de convergence faible du type «Wong-Zakai» quand les approximants sont reguliers et continus, meme si les limites ne sont pas continues

Stochastic Integrals: An Introduction

Abstract: This text introduces at a moderate speed and in a thorough way the basic concepts of the theory of stochastic integrals and Ito calculus for sem i martingales. There are many reasons to study this subject. We are fascinated by the contrast between general measure theoretic arguments and concrete probabilistic problems, and by the own flavour of a new differential calculus. For the beginner, a lot of work is necessary to go through this text in detail. As areward it should enable her or hirn to study more advanced literature and to become at ease with a couple of seemingly frightening concepts. Already in this introduction, many enjoyable and useful facets of stochastic analysis show up. We start out having a glance at several elementary predecessors of the stochastic integral and sketching some ideas behind the abstract theory of semimartingale integration. Having introduced martingales and local martingales in chapters 2 - 4, the stochastic integral is defined for locally uniform limits of elementary processes in chapter S. This corresponds to the Riemann integral in one-dimensional analysis and it suffices for the study of Brownian motion and diffusion processes in the later chapters 9 and 12.

Itô correction terms for the radial parts of semimartingales on manifolds

Abstract: We consider functions,F, of a semimartingale,X, on a complete manifold which fail to beC2 only on, and are sufficiently well-behaved near, a codimension 1 subset ℒ. We obtain an extension of the Ito formula which is valid for all time by adding a continuous predictable process given explicitly in terms of two geometric local times ofX on ℒ and the Gâteaux derivative ofF. We then examine the cut locus of a point of the manifold in sufficient detail to show that this result applies to give a corresponding expression for the radial part of the semimartingale.

Exponential stability of stochastic differential equations driven by discontinuous semimartingales

Abstract: Both almost sure exponential stability and exponential stability in mean square for stochastic differential equations driven by general semimartingales, which may not be continuous, are investigated under various hypotheses. New methods are introduced to incorporate the discontinuity of semimartingales. The Doleans Dade stochastic exponential, the convergence of nonnegative special semimartingales established by Liptser & Shiryayey [5] as well as the lto formula will play a great role in this paper

Local times and almost sure convergence of semi-martingales

Abstract: © Gauthier-Villars, 1995, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section B » (http://www.elsevier.com/locate/anihpb) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

The probabilities of large deviations for semimartingales

Abstract: Let be the square integrabie martingale with the quadratic characteristic be the standard normal d.f.. Assume that there are real numbers EeLand EeCsuch that the following conditions hold a.s. Denote where . Then for any rin the range the large deviation relation holds where is a constant depending only on cand

The Linear-Quadratic Control Problem Revisited

Abstract: A long-run, average-cost, stochastic, linear-quadratic control problem that incorporates different time scales is considered. The system dynamics and the cost functional are modeled with the help of a locally square-integrable semimartingale process with independent increments and the corresponding predictable quadratic variation process. The solution of the control problem is given in terms of the solution of certain system of algebraic and differential Riccati equations. The model considered here embodies as particular cases the "traditional" continuous-time and discrete-time linear quadratic control problems, and is applicable, for example, to certain hybrid control problems that cannot be treated using existing control methods.

A Fubini Theorem for Generalized Stratonovich Integrals

Abstract: The purpose of this paper is to analyze under which conditions the multiple generalized (that means, non necessarily adapted) Stratonovich integral with respect to the Brownian sheet can be iterated. The motivation of this problem comes from a previous work by the authors. Indeed, in [2] an iteration of a double Stratonovich integral with respect to a non adapted “semimartingale” in the plane is needed in order to prove a Green formula. In comparison with that article the situation considered here is more simple, since our integrator is the Brownian sheet, but also more general, because we are considering multiple integrals of any order k ≥ 2. The basic ingredients which are needed are the Hu-Meyer formula established in [1], the Fubini’s theorem for the multiple Skorohod integral (see [3]) and some results on the iteration of traces.

Reflected Brownian motion in a cone: Semimartingale property*

Abstract: In this paper, the object of study is reflected Brownian motion in a cone ind-dimensions (d≧3) with nonconstant oblique reflection on each radial line emanating from the vertex of the cone The basic question considered here is “When is this process a semimartingale?” Conditions for the existence and uniqueness of the process for which the vertex is an instantaneous state were given by Kwon, which is resolved in terms of a real parameter α∞ depending on the cone and the direction of reflection It is shown that starting from any point of the cone, the process is a semimartingale if α∞ < 1, α∞ + 0 and not a semimartingale if < α∞ < 2

Martingale decompositions and integration

Abstract: Continuing the work of the preceding chapter on continuous parameter (sub)martingales, we present a solution of the Doob decomposition problem, raised in Section 2.5, which is due to Meyer. We give an elementary (but longer) demonstration and also sketch a shorter (but more sophisticated) argument based on Doleans-Dade signed measure representation of quasimartingales. This decomposition leads to stochastic integration with square integrable martingales as integrators generalizing the classical Ito integration. The material is presented in this chapter in considerable detail, since it forms a basis for semimartingale integrals with numerous applications to be abstracted and treated in the following chapter. Orthogonal decompositions of square integrable martingales (of continuous time parameter), its time change transformation leading to a related Brownian motion process and the Levy characterization of the latter from continuous parameter martingales, are covered. Stopping (or optional) times play a key role in all this work, and some classifications of these are given. The treatment also includes the Stratonovich integrals as well as an identification of the square integrable martingale integrators with spectral measures of certain normal operators in Hilbert space. Finally some related results appear as exercises in the Complements section.