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Replication of Wiener-Transformable Stochastic Processes with Application to Financial Markets with Memory

  • Elena Boguslavskaya
  • Yuliya Mishura
  • Georgiy Shevchenko
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 271)

Abstract

We investigate Wiener-transformable markets, where the driving process is given by an adapted transformation of a Wiener process. This includes processes with long memory, like fractional Brownian motion and related processes, and, in general, Gaussian processes satisfying certain regularity conditions on their covariance functions. Our choice of markets is motivated by the well-known phenomena of the so-called “constant” and “variable depth” memory observed in real world price processes, for which fractional and multifractional models are the most adequate descriptions. Motivated by integral representation results in general Gaussian setting, we study the conditions under which random variables can be represented as pathwise integrals with respect to the driving process. From financial point of view, it means that we give the conditions of replication of contingent claims on such markets. As an application of our results, we consider the utility maximization problem in our specific setting. Note that the markets under consideration can be both arbitrage and arbitrage-free, and moreover, we give the representation results in terms of bounded strategies.

Keywords

Wiener-transformable process Fractional Brownian motion Long memory Pathwise integral Martingale representation Utility maximization 

Notes

Acknowledgements

Elena Boguslavskaya is supported by Daphne Jackson fellowship funded by EPSRC. The research of Yu. Mishura was funded (partially) by the Australian Government through the Australian Research Council (project number DP150102758). Yu. Mishura acknowledges that the present research is carried through within the frame and support of the ToppForsk project nr. 274410 of the Research Council of Norway with title STORM: Stochastics for Time-Space Risk Models.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Elena Boguslavskaya
    • 1
  • Yuliya Mishura
    • 2
  • Georgiy Shevchenko
    • 2
  1. 1.Department of MathematicsBrunel University LondonUxbridgeUK
  2. 2.Department of Probability Theory, Statistics and Actuarial MathematicsTaras Shevchenko National University of KyivKyivUkraine

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