A dynamical thermostat approach to financial asset price dynamics

Thurner, Stefan

doi:10.1063/1.1386819

Cited by 1 publication

(1 citation statement)

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“…Grassia has also studied a similar linear second-order differential equation derived from market delay, positive feedback and including a mechanism for quenching runaway markets [19]. Thurner [55] considers a three-dimensional system of three ordinary differential equations coupling price, "friction" and a state variable controlling friction, which can be mapped onto a third-order ordinary differential equation. The nonlinearity is on the friction term and not on the trend term which is again assumed linear.…”

Section: Nonlinear Dynamical Equation For Stock Market Pricesmentioning

confidence: 99%

Oscillatory finite-time singularities in finance, population and rupture

Ide

Sornette

2002

Physica A: Statistical Mechanics and its Applications

133

155

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We present a simple two-dimensional dynamical system where two nonlinear terms, exerting respectively positive feedback and reversal, compete to create a singularity in finite time decorated by accelerating oscillations. The power law singularity results from the increasing growth rate. The oscillations result from the restoring mechanism. As a function of the order of the nonlinearity of the growth rate and of the restoring term, a rich variety of behavior is documented analytically and numerically. The dynamical behavior is traced back fundamentally to the self-similar spiral structure of trajectories in phase space unfolding around an unstable spiral point at the origin. The interplay between the restoring mechanism and the nonlinear growth rate leads to approximately log-periodic oscillations with remarkable scaling properties. Three domains of applications are discussed: (1) the stock market with a competition between nonlinear trend-followers and nonlinear value investors; (2) the world human population with a competition between a population-dependent growth rate and a nonlinear dependence on a finite carrying capacity;(3) the failure of a material subjected to a time-varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing. 7 Overall dynamics for n > 1 and m > 2 with α > 0: |y 1 (t → t c )| < +∞ (except for isolated initial conditions) and |y 2 (t → t c )| = +∞ Recall from Section 5.1 for the sub-dynamical system with only the oscillatory element that the case n > 1 corresponds to highly nonlinear oscillations with a monotonically decreasing period as the amplitude of the oscillations increases ( Figure 2). From Section 5.2 for the sub-dynamical system with only the singular element, the case m > 2 corresponds to finite-time singularity with finite increment in y 1 and infinite increment in y 2 (Figure 4). Furthermore, Section 6.3 on the phase space of the full dynamical system showed the following results:1. any trajectory y(t; y 0 , t 0 ) starting away from the origin connects the origin in backward time and |y 2 | → ∞ in forward time;

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Section: Nonlinear Dynamical Equation For Stock Market Pricesmentioning

confidence: 99%