Dynamics of Moving Average Rules in a Continuous-time Financial Market Model

Within a continuous-time framework, this paper proposes a stochastic heterogeneous agent model (HAM) of financial markets with time delays to unify various moving average rules used indiscrete-time HAMs. The time delay represents a memory length of a moving average rule indiscrete-time HAMs.Intuitive conditions for the stability of the fundamental price of the deterministic model in terms of agents' behavior parameters and memory length are obtained. It is found that an increase in memory length not only can destabilize the market price, resulting in oscillatory market price characterized by a Hopf bifurcation, but also can stabilize another wise unstable market price, leading to stability switching as the memory length increases. Numerical simulations show that the stochastic model is able to characterize long deviations of the market price from its fundamental price and excess volatility and generate most of the stylized factso bserved in financial markets.asset price; financial market behavior; heterogeneous beliefs; stochastic delay differential equations; stability; bifurcations; stylized facts

Time-Dependent Scalar Fields in Modified Gravities in a Stationary Spacetime

Most no-hair theorems involve the assumption that the scalar field is independent of time. Recently in [Phys. Rev. D90 (2014) 041501(R)] the existence of time-dependent scalar hair outside a stationary black hole in general relativity was ruled out. We generalize this work to modified gravities and non-minimally coupled scalar field with an additional assumption that the spacetime is axisymmetric. It is shown that in higher-order gravity such as metric $f(R)$ gravity the time-dependent scalar hair doesn't exist. While in Palatini $f(R)$ gravity and non-minimally coupled case the time-dependent scalar hair may exist.Comment: 6 pages, no figure

The Stochastic Dynamics of Speculative Prices

Within the framework of the heterogeneous agent paradigm, we establish a stochastic model of speculative price dynamics involving of two types of agents, fundamentalists and chartists, and the market price equilibria of which can be characterised by the invariant measures of a random dynamical system. By conducting a stochastic bifurcation analysis, we examine the market impact of speculative behaviour. We show that, when the chartists use lagged price trends to form their expectations, the market equilibrium price can be characterised by a unique and stable invariant measure when the activity of the speculators is below a certain critical value. If this threshold is surpassed, the market equilibrium can be characterised by more than two invariant measures, of which one is completely stable, another is completely unstable and the remaining ones may exhibit various types of stability. Also, the corresponding stationary measure displays a significant qualitative change near the threshold value. We show that the stochastic model displays behaviour consistent with that of the underlying deterministic model. However, when the time lag in the formation of the price trends used by the chartists approaches zero, such consistency breaks down. In addition, the change in the stationary distribution is consistent with a number of market anomalies and stylised facts observed in financial markets, including a bimodal logarithmic price distribution and fat tails.heterogeneous agents; speculative behaviour; random dynamical systems; stochastic bifurcations; invariant measures; chartists

Abnormal Synchronizing Path of Delay-coupled Chaotic Oscillators on the Edge of Stability

In this paper, the transition of synchronizing path of delay-coupled chaotic oscillators in a scale-free network is highlighted. Mainly, through the critical transmission delay makes chaotic oscillators be coupled on the edge of stability, we find that the transition of synchronizing path is \emph{abnormal}, which is characterized by the following evidences: (a) synchronization process starts with low-degree rather than high-degree ones; (b) the high-degree nodes don't undertake the role of hub; (c) the synchronized subnetworks show a poor small-world property as a result of hubs absence; (d) the clustering synchronization behavior emerges even community structure is absent in the scale-free network. This abnormal synchronizing path suggests that the diverse synchronization behaviors occur in the same topology, which implies that the relationship between dynamics and structure of network is much more complicated than the common sense that the structure is the foundation of dynamics. Moreover, it also reveals the potential connection from the transition of synchronization behavior to disorder in real complex networks, e.g. Alzheimer disease.Comment: 12 pages, 4 figure