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4 results

  • On mixed fractional stochastic differential equations with discontinuous drift coefficient

  • Part of
  • Ercan Sönmez
  • Journal:
    Journal of Applied Probability / Volume 60 / Issue 2 / June 2023
    Published online by Cambridge University Press:
    01 December 2022, pp. 589-606
    Print publication:
    June 2023
    • Article
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  • We prove existence and uniqueness for the solution of a class of mixed fractional stochastic differential equations with discontinuous drift driven by both standard and fractional Brownian motion. Additionally, we establish a generalized Itô rule valid for functions with an absolutely continuous derivative and applicable to solutions of mixed fractional stochastic differential equations with Lipschitz coefficients, which plays a key role in our proof of existence and uniqueness. The proof of such a formula is new and relies on showing the existence of a density of the law under mild assumptions on the diffusion coefficient.

  • 2 - Stochastic Integrals and Differentials

  • from Part I - Point Processes
  • Tomas Björk, Stockholm School of Economics
  • Book:
    Point Processes and Jump Diffusions
    Published online:
    27 May 2021
    Print publication:
    17 June 2021, pp 8-29

  • Summary

    The necessary stochastic integral theory for processes of bounded variation is developed. The concept of predictability is discussed in detail, its interpretation as well as its relation to martingale theory. The Itô formula is derived and we prove the Watanabe characterization of the Poisson process.

  • 8 - The Itô Formula

  • from Part I - Point Processes
  • Tomas Björk, Stockholm School of Economics
  • Book:
    Point Processes and Jump Diffusions
    Published online:
    27 May 2021
    Print publication:
    17 June 2021, pp 72-81

  • Summary

    We derive the relevant Itô formula for jump diffusions driven by a marked point process. The compound Poisson process is introduced, and we also study the MPP versions of the linear and geometric SDE.

Point Processes and Jump Diffusions
  • An Introduction with Finance Applications
  • Tomas Björk
  • Published online:
    27 May 2021
    Print publication:
    17 June 2021
  • The theory of marked point processes on the real line is of great and increasing importance in areas such as insurance mathematics, queuing theory and financial economics. However, the theory is often viewed as technically and conceptually difficult and has proved to be a block for PhD students looking to enter the area. This book gives an intuitive picture of the central concepts as well as the deeper results, while presenting the mathematical theory in a rigorous fashion and discussing applications in filtering theory and financial economics. Consequently, readers will get a deep understanding of the theory and how to use it. A number of exercises of differing levels of difficulty are included, providing opportunities to put new ideas into practice. Graduate students in mathematics, finance and economics will gain a good working knowledge of point-process theory, allowing them to progress to independent research.