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.
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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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Boguslavskaya, E., Mishura, Y., Shevchenko, G. (2018). Replication of Wiener-Transformable Stochastic Processes with Application to Financial Markets with Memory. In: Silvestrov, S., Malyarenko, A., Rančić, M. (eds) Stochastic Processes and Applications. SPAS 2017. Springer Proceedings in Mathematics & Statistics, vol 271. Springer, Cham. https://doi.org/10.1007/978-3-030-02825-1_14
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