© 2012

Sharp Martingale and Semimartingale Inequalities

  • Aims at a detailed explanation of Burkholder's method: presents, for most estimates, the steps leading to the discovery of the corresponding special functions

  • Uses diverse analytic and probabilistic methods to solve the corresponding boundary value problems

  • Presents a unified up-to-date treatment, illustrated on a variety of examples of different type, difficulty and complexity

  • Material is completely self-contained


Part of the Monografie Matematyczne book series (MONOGRAFIE, volume 72)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Adam Osękowski
    Pages 1-7
  3. Adam Osękowski
    Pages 9-22
  4. Adam Osękowski
    Pages 23-141
  5. Adam Osękowski
    Pages 211-243
  6. Adam Osękowski
    Pages 245-293
  7. Adam Osękowski
    Pages 295-392
  8. Adam Osękowski
    Pages 393-446
  9. Back Matter
    Pages 447-462

About this book


This monograph presents a unified approach to a certain class of semimartingale inequalities, which can be regarded as probabilistic extensions of classical estimates for conjugate harmonic functions on the unit disc. The approach, which has its roots in the seminal works of Burkholder in the 1980s, makes it possible to deduce a given inequality for semimartingales from the existence of a certain special function with some convex-type properties. Remarkably, an appropriate application of the method leads to the sharp version of the estimate under investigation, which is particularly important for applications. These include the theory of quasiregular mappings (with major implications for the geometric function theory); the boundedness of two-dimensional Hilbert transforms and a more general class of Fourier multipliers; the theory of rank-one convex and quasiconvex functions; and more.

The book is divided into a number of distinct parts. In the introductory chapter we present the motivation for the results and relate them to some classical problems in harmonic analysis. The next part contains a general description of the method, which is applied in subsequent chapters to the study of sharp estimates for discrete-time martingales; discrete-time sub- and supermartingales; continuous time processes; and the square and maximal functions. Each chapter contains additional bibliographical notes included for reference purposes.


best constants differential subordination inequality martingale semimartingale

Authors and affiliations

  1. 1., Institute of MathematicsUniversity of WarsawWarsawPoland

Bibliographic information

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