An analytic formulation of memory-possessing random walks introduced recently [Cressoni et al., Phys. Rev. Lett. 98, 070603 (2007) and Sch\"utz and Trimper, Phys. Rev. E 70, 045101 (2004)] for Alzheimer behavior and related phenomena is provided along with exact solutions on the basis of Fokker-Planck equations. The solution of a delay-differential equation derived for the purpose is shown to produce log-periodic oscillations and to coincide rather accurately with previously published computer simulation results. Generalizations along several directions are also constructed on the basis of the formalism.
Two remarkable publications have recently appeared on the subject of random walks with memory, one in- volving the ‘elephant walk’ in which the walker chooses steps randomly but is influenced by a perfect memory of steps taken earlier [1], and the other involving an extension of this walk to incorporate partial memory of steps from the beginning up to a time in the past [2]. While an analytical description has been given for the (former) elephant walk, it appears to have been impossible to provide one for the partial memory extension. The significance of the latter is that it has been proposed [2] for the medically important analysis of amnestically in- duced behavior of Alzheimer patients. The authors of Ref. [2] have presented impressive computer simulations of the Alzheimer walk exhibiting log-periodic oscillations in the displacement of the walker, and deduced intriguing conclusions regarding the elements of persistence and what they have called, following Schu ̈tz and Trimper [1], traditional versus reformer behavior of the walker. They have also stated that an analytic solution of their partial memory extension (the Alzheimer walk) remains an open problem. The present Letter is aimed at solving that problem.The quote above is from an interesting paper on random walks with either perfect memory (elephant walks) or memory of the past without recent memory (Alzheimer's walks). The latter is given an analytic treatment based on a time dependent Fokker Planck Equation. This is of interest for two reasons: 1. our Coherent Market Hypothesis (CMH) is based on the Fokker Planck Equation; and 2. The log periodic oscillations that occur with Alzheimer's random walks have empirically been found to be precursors to market bubbles and crashes (or at least regime changes) by Didier Sornette, et. al.
The CMH is based on a stationary (time independent) Fokker Planck Equation. Therefore this paper by V. M. Kenkre paves the way for introducing time dependence into the CMH formalism leading to prediction of log periodic oscillations in the financial markets. Clearly this is an important opportunity for research on extending the CMH into time dependent as well as stationary state dynamics.
