Replication of Wiener-Transformable Stochastic Processes with Application to Financial Markets with Memory
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.
KeywordsWiener-transformable process Fractional Brownian motion Long memory Pathwise integral Martingale representation Utility maximization
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.
- 4.Björk, T.: Arbitrage Theory in Continuous Time, 2nd edn. Oxford University Press, Oxford (2004)Google Scholar
- 9.Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. Translated from the French, Studies in Mathematics and its Applications, vol. 1. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York (1976)Google Scholar
- 10.Föllmer, H., Schied, A.: Stochastic Finance. An Introduction in Discrete Time, Extended edn. Walter de Gruyter & Co., Berlin (2011)Google Scholar
- 11.Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, vol. 113, 2nd edn. Springer, New York (1991)Google Scholar
- 13.Li, W.V., Shao, Q.M.: Gaussian processes: inequalities, small ball probabilities and applications. In: Handbook of Statistics, vol. 19, pp. 533–597 (2001)Google Scholar
- 22.Shevchenko, G., Viitasaari, L.: Adapted integral representations of random variables. Int. J. Modern Phys.: Conf. Ser. 36, Article ID 1560004, 16 (2015)Google Scholar
- 23.Zähle, M.: On the link between fractional and stochastic calculus. In: Stochastic Dynamics (Bremen, 1997), pp. 305–325. Springer, New York (1999)Google Scholar