Mathematica Bohemica, Vol. 141, No. 2, pp. 261-286, 2016
The Kurzweil integral in financial market modeling
Pavel Krejčí, Harbir Lamba, Giselle Antunes Monteiro, Dmitrii Rachinskii
Abstract: 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.
Keywords: hysteresis; Prandtl-Ishlinskii operator; Kurzweil integral; market model
Affiliations: Pavel Krejčí, Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic, e-mail: krejci@math.cas.cz; Harbir Lamba, Department of Mathematical Sciences, George Mason University, 4459 Exploratory Hall, Rivanna River Lane, Fairfax, Virginia 22030, USA, e-mail: hlamba@gmu.edu; Giselle Antunes Monteiro, Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic, e-mail: gam@math.cas.cz; Dmitrii Rachinskii, Department of Mathematical Sciences, University of Texas at Dallas, 800 W. Campbell Road, TX 75080-3021 Richardson, Texas 75080, USA, e-mail: dmitry.rachinskiy@utdallas.edu
