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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Androshchuk, T., Mishura, Y.: Mixed Brownian-fractional Brownian model: absence of arbitrage and related topics. Stochastics 78(5), 281–300 (2006)
Bender, C., Sottinen, T., Valkeila, E.: Pricing by hedging and no-arbitrage beyond semimartingales. Financ. Stoch. 12(4), 441–468 (2008)
Bender, C., Sottinen, T., Valkeila, E.: Fractional processes as models in stochastic finance. In: Advanced Mathematical Methods for Finance, pp. 75–103. Springer, Heidelberg (2011)
Björk, T.: Arbitrage Theory in Continuous Time, 2nd edn. Oxford University Press, Oxford (2004)
Cheridito, P.: Mixed fractional Brownian motion. Bernoulli 7(6), 913–934 (2001)
Cheridito, P.: Arbitrage in fractional Brownian motion models. Financ. Stoch. 7(4), 533–553 (2003)
Dudley, R.M.: Wiener functionals as Itô integrals. Ann. Probab. 5(1), 140–141 (1977)
Dung, N.T.: Semimartingale approximation of fractional Brownian motion and its applications. Comput. Math. Appl. 61(7), 1844–1854 (2011)
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)
Föllmer, H., Schied, A.: Stochastic Finance. An Introduction in Discrete Time, Extended edn. Walter de Gruyter & Co., Berlin (2011)
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, vol. 113, 2nd edn. Springer, New York (1991)
Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance, Applications of Mathematics (New York), vol. 39. Springer, New York (1998)
Li, W.V., Shao, Q.M.: Gaussian processes: inequalities, small ball probabilities and applications. In: Handbook of Statistics, vol. 19, pp. 533–597 (2001)
Lifshits, M.A.: Gaussian Random Functions, Mathematics and its Applications, vol. 322. Kluwer Academic Publishers, Dordrecht (1995)
Mishura, Y., Shevchenko, G.: Small ball properties and representation results. Stoch. Process. Appl. 127(1), 20–36 (2017)
Mishura, Y., Shevchenko, G., Valkeila, E.: Random variables as pathwise integrals with respect to fractional Brownian motion. Stoch. Process. Appl. 123(6), 2353–2369 (2013)
Mishura, Y.S.: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Mathematics, vol. 1929. Springer, Berlin (2008)
Norros, I., Valkeila, E., Virtamo, J.: An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5(4), 571–587 (1999)
Rogers, L.C.G.: Arbitrage with fractional Brownian motion. Math. Financ. 7(1), 95–105 (1997)
Samko, S.G., Ross, B.: Integration and differentiation to a variable fractional order. Integral Transform. Spec. Funct. 1(4), 277–300 (1993)
Shalaiko, T., Shevchenko, G.: Integral representation with respect to fractional brownian motion under a log-hölder assumption. Modern Stoch.: Theory Appl. 2(3), 219–232 (2015)
Shevchenko, G., Viitasaari, L.: Adapted integral representations of random variables. Int. J. Modern Phys.: Conf. Ser. 36, Article ID 1560004, 16 (2015)
Zähle, M.: On the link between fractional and stochastic calculus. In: Stochastic Dynamics (Bremen, 1997), pp. 305–325. Springer, New York (1999)
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-030-02825-1_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-02824-4
Online ISBN: 978-3-030-02825-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)