The Kurzweil integral in financial market modeling. (English) Zbl 1389.34140

Krejčí, Pavel; Lamba, Harbir; Monteiro, Giselle Antunes; Rachinskii, Dmitrii

Math. Bohem. 141, No. 2, 261-286 (2016).

Summary: Certain financial market strategies are known to exhibit a hysteretic structure similar to the memory observed in plasticity, ferromagnetism, or magnetostriction. The main difference is that in financial markets, the spontaneous occurrence of discontinuities in the time evolution has to be taken into account. We show that one particular market model considered here admits a representation in terms of Prandtl-Ishlinskii hysteresis operators, which are extended in order to include possible discontinuities both in time and in memory. The main analytical tool is the Kurzweil integral formalism, and the main result proves the well-posedness of the process in the space of right-continuous regulated functions.

MSC:

34C60	Qualitative investigation and simulation of ordinary differential equation models
34C55	Hysteresis for ordinary differential equations
26A39	Denjoy and Perron integrals, other special integrals
91B26	Auctions, bargaining, bidding and selling, and other market models

Keywords:

hysteresis; Prandtl-Ishlinskii operator; Kurzweil integral; market model