Topic
Semimartingale
About: Semimartingale is a research topic. Over the lifetime, 1497 publications have been published within this topic receiving 42406 citations.
Papers published on a yearly basis
Papers
More filters
Book•
01 Jan 1987
TL;DR: In this article, the General Theory of Stochastic Processes, Semimartingales, and Stochastically Integrals is discussed and the convergence of Processes with Independent Increments is discussed.
Abstract: I. The General Theory of Stochastic Processes, Semimartingales and Stochastic Integrals.- II. Characteristics of Semimartingales and Processes with Independent Increments.- III. Martingale Problems and Changes of Measures.- IV. Hellinger Processes, Absolute Continuity and Singularity of Measures.- V. Contiguity, Entire Separation, Convergence in Variation.- VI. Skorokhod Topology and Convergence of Processes.- VII. Convergence of Processes with Independent Increments.- VIII. Convergence to a Process with Independent Increments.- IX. Convergence to a Semimartingale.- X. Limit Theorems, Density Processes and Contiguity.- Bibliographical Comments.- References.- Index of Symbols.- Index of Terminology.- Index of Topics.- Index of Conditions for Limit Theorems.
5,987 citations
TL;DR: In this paper, a general stochastic model of a frictionless security market with continuous trading is developed, where the vector price process is given by a semimartingale of a certain class, and the general Stochastic integral is used to represent capital gains.
2,851 citations
Book•
01 Jan 1989
TL;DR: In this article, a review of results of the General Theory of Stochastic Processes is presented, with a focus on the weak convergence of distributions of semimartingales to distributions of point processes.
Abstract: I.- 1. Basic Concepts and the Review of Results of "The General Theory of Stochastic Processes".- 1. Stochastic basis. Random times, sets and processes.- 2. Optional and predictable ?-algebras of random sets.- 3. Predictable and totally inaccessible random times. Classification of Markov times. Section theorems.- 4. Martingales and local martingales.- 5. Square integrable martingales.- 6. Increasing processes. Compensators (dual predictable projections). The Doob-Meyer decomposition.- 7. The structure of local martingales.- 8. Quadratic characteristic and quadratic variation.- 9. Inequalities for local martingales.- 2. Semimartingales. I. Stochastic Integral.- 1. Semimartingales and quasimartingales.- 2. Stochastic integral with respect to a local martingale and a semimartingale. Construction and properties.- 3. Ito's formula. I.- 4. Doleans equation. Stochastic exponential.- 5. Multiplicative decomposition of positive semimartingales.- 6. Convergence sets and the strong law of large numbers for special martingales.- 3. Random Measures and their Compensators.- 1. Optional and predictable random measures.- 2. Compensators of random measures. Conditional mathematical expectation with respect to the ?-algebra P?.- 3. Integer-valued random measures.- 4. Multivariate point processes.- 5. Stochastic integral with respect to a martingale measure ?-?.- 6. Ito's formula. II.- 4. Semimartingales. II Canonical Representation.- 1. Canonical representation. Triplet of predictable characteristics of a semimartingale.- 2. Stochastic exponential constructed by the triplet of a semimartingale.- 3. Martingale characterization of semimartingales by means of stochastic exponentials.- 4. Characterization of semimartingales with conditionally independent increments.- 5. Semimartingales and change of probability measures. Transformation of triplets.- 6. Semimartingales and reduction of a flow of ?-algebras.- 7. Semimartingales and random change of time.- 8. Semimartingales and integral representation of martingales.- 9. Gaussian martingales and semimartingales.- 10. Filtration of special semimartingales.- 11. Semimartingales and helices. Ergodic theorems.- 12. Semimartingales - stationary processes.- 13. Exponential inequalities for large deviation probabilities.- II.- 5. Weak Convergence of Finite-Dimensional Distributions of Semimartingales to Distributions of Processes with Conditionally Independent Increments.- 1. Method of stochastic exponentials. I. Convergence of conditional characteristic functions.- 2. Method of stochastic exponentials. II. Weak convergence of finite dimensional distributions.- 3. Weak convergence of finite dimensional distributions of point processes and semimartingales to distributions of point processes.- 4. Weak convergence of finite dimensional distributions of semimartingales to distributions of a left quasi-continuous semimartingale with conditionally independent increments.- 5. The central limit theorem. I. "Classical" version.- 6. The central limit theorem. II. "Nonclassical" version.- 7. Evaluation of a convergence rate for marginal distributions in the central limit theorem.- 8. A martingale method of proving the central limit theorem for strictly stationary sequences. Relation to mixing conditions.- 6. The Space D. Relative Compactness of Probability Distributions of Semimartingales.- 1. The space D. Skorohod's topology.- 2. Continuous functions on R+ x D.- 3. Conditions on adapted processes sufficient for relative compactness of families of their distributions.- 4. Relative compactness of probability distributions of semimartingales.- 5. Conditions necessary for the weak convergence of probability distributions of semimartingales.- 7. Weak Convergence of Distributions of Semimartingales to Distributions of Processes with Conditionally Independent Increments.- 1. The functional central limit theorem (invariance principle).- 2. Weak convergence of distributions of semimartingales to distributions of point processes.- 3. Weak convergence of distributions of semimartingales to the distribution of a left quasi-continuous semimartingale, with conditionally independent increments.- 8. Weak Convergence of Distributions of Semimartingales to the Distribution of a Semimartingale.- 1. Convergence of stochastic exponentials and weak convergence of distributions of semimartingales.- 2. Weak convergence to the distribution of a left quasi-continuous semimartingale.- 3. Diffusion approximation.- 4. Weak convergence to a distribution of a point process with a continuous compensator.- 5. Weak convergence of in variant measures.- III.- 9. Invariance Principle and Diffusion Approximation for Models Generated by Stationary Processes.- 1. Generalization of Donsker's invariance principle.- 2. Invariance principle for strictly stationary processes.- 3. Invariance principle for a Markov process.- 4. Diffusion approximation for systems with a "broad bandwidth noise" (scalar case).- 5. Diffusion approximation with a "broad bandwidth noise" (vector case).- 6. Ergodic theorem and invariant principle in case of nonhomogeneous time averaging.- 7. Stochastic version of Bogoljubov's averaging principle.- 10. Diffusion Approximation for Semimartingales with a Normal Reflexion in a Convex Region.- 1. Skorohod's problem on normal reflection.- 2. Semimartingale with normal reflection.- 3. Diffusion approximation with normal reflection.- 4. Diffusion approximation with reflection for queueing models with autonomious service.- Historic-Bibliographical notes.
871 citations
Book•
12 Mar 2014
TL;DR: In this article, the authors present a rapid introduction to the modern semimartingale theory of stochastic integration and differential equations, without first having to treat the beautiful but highly technical "general theory of processes".
Abstract: This book is quite different from others on the subject in that it presents a rapid introduction to the modern semimartingale theory of stochastic integration and differential equations, without first having to treat the beautiful but highly technical "general theory of processes". The author's new approach (based on the theorem of Bitcheler-Dellacherie) also give a more intuitive understanding of the subject, and permits proofs to be much less technical. All of the major theorems of stochastic integration are given, including a comprehensive treatment (first time in English) of local times. A theory of stochastic differential equations driven by semimartingales is developed, including Fisk-Stratonovich equations, Markov properties, stability, and an introduction to the theory of flows. Further topics presented for the 1st time in book form include an elementary presentation of Azema's martingale. This book will quickly become a standard reference on the subject, to be used by specialists and non-specialists alike, both for the sake of the theory and for its application.
847 citations
TL;DR: In this paper, the authors show that the RV is sometimes a quite noisy estimator of integrated variance, even with large values of M. The authors use the limit theory on some exchange rate data and some stock data.
Abstract: This paper looks at some recent work on estimating quadratic variation using realised variance (RV) — that is sums of M squared returns. This econometrics has been motivated by the advent of the common availability of high frequency financial return data. When the underlying process is a semimartingale we recall the fundamental result that RV is a consistent (as M →∞ ) estimator of quadratic variation (QV). We express concern that without additional assumptions it seems difficult to give any measure of uncertainty of the RV in this context. The position dramatically changes when we work with a rather general SV model — which is a special case of the semimartingale model. Then QV is integrated variance and we can derive the asymptotic distribution of the RV and its rate of convergence. These results do not require us to specify a model for either the drift or volatility functions, although we have to impose some weak regularity assumptions. We illustrate the use of the limit theory on some exchange rate data and some stock data. We show that even with large values of M the RV is sometimes a quite noisy estimator of integrated variance.
626 citations