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This paper establishes a continuous-time stochastic asset pricing model in a speculative financial market with fundamentalists and chartists by introducing a noisy fundamental price. By application of stochastic bifurcation theory, the limiting market equilibrium distribution is examined numerically. It is shown that speculative behaviour of chartists can cause the market price to display different forms of equilibrium distributions. In particular, when chartists are less active, there is a unique equilibrium distribution which is stable. However, when the chartists become more active, a new equilibrium distribution will be generated and become stable. The corresponding stationary density will change from a single peak to a crater-like density. The change of stationary distribution is characterized by a bimodal logarithm price distribution and fat tails. The paper demonstrates that stochastic bifurcation theory is a useful tool in providing insight into various types of financial market behaviour in a stochastic environment.

The theory of random dynamical systems provides a very powerful mathematical tool for understanding the limiting behaviour of stochastic systems. Recently, it has been applied to economics and finance to help in understanding the stochastic nature of financial models with random perturbations. In particular, the study of the limiting distribution of various stochastic models in economics and finance gives a good description of stationary markets. For example, Föllmer et al. [9] study the existence and uniqueness of the limiting distribution in a discrete financial market model with different expectations through stochastic learning and Böhm and Chiarella [4] investigate the long-run behaviour (stationary solutions) for mean-variance preferences under various predictors. Those models mainly focus on the existence, uniqueness and stability of limiting distributions of discrete-time financial models, rather than the changes of existence and stability of multiple limiting distributions of continuous-time financial models as their parameters change. Stochastic bifurcation theory has been developed to study the changes of existence and stability of limiting distributions. There seems to have been no application of it to heterogeneous agent models of continuous-time financial markets except the one-dimensional continuously randomized version of Zeeman’s [15] stock market model studied by Rheinlaender and Steinkamp [12]. These authors show a stochastic stabilization effect and possible sudden trend reversal in the financial market. For higher dimensional financial market models with heterogeneous agents in the continuous-time framework, the application of stochastic bifurcation theory faces many challenges. In this paper, we take the very basic heterogeneous agent financial market model of fundamentalists and chartists developed by Beja and Goldman [3] and Chiarella [5] and set it up as a two-dimensional stochastic model by introducing a noisy fundamental price in a continuous-time framework. We then use stochastic bifurcation theory to analyze numerically the changes and stability of multiple limiting distributions of the two-dimensional financial market model as the chartists’ behaviour changes. The numerical analysis of our speculative market model is largely motivated by the work of Arnold et al. [2] and Schenk-Hoppé [14] on the noisy Duffing–van der Pol oscillator. To provide a complete picture of the market equilibrium behaviour of the model as a parameter capturing the chartists behaviour changes, we conduct our analysis from the viewpoints of both dynamical and phenomenological bifurcations. The so-called dynamical (D)-bifurcation examines the evolution of an initial point forward and backward in time and captures all the stochastic dynamics of the SDEs, while the so-called phenomenological (P)-bifurcation studies a stationary measure corresponding to the one-point motion. As indicated in Schenk-Hoppé [14] and the references cited there, the difference between P-bifurcation and D-bifurcation lies in the fact that the P-bifurcation approach focuses on long-run probability distributions, while the D-bifurcation approach is based on the invariant measure, the multiplicative ergodic theorem, the Lyapunov exponents and the occurrence of new invariant measures. However, the P-bifurcation has the advantage of allowing one to visualize the changes of the stationary density functions. Our results show that, as the chartists change their behaviour (through their extrapolation of the price trend), the market price can display different forms of equilibrium distribution. In particular, when chartists are less active, the market has a unique equilibrium distribution which is stable. However, when the chartists become more active, a new equilibrium distribution will be generated and become stable whilst the original equilibrium distribution becomes unstable. The corresponding stationary density will change from a single peak to a crater-like density. The market price can be driven away from the fundamental price. The change of stationary distribution is characterized by a bimodal logarithm price distribution and fat tails. The structure of the paper is as follows. In Section 2, we first outline the extended model of Beja and Goldman [3] and Chiarella [5] with a stochastic fundamental price. In Sections 3 and 4, the stochastic dynamical behaviour is analyzed from the viewpoints of invariant measures and stationary measures respectively. The paper is then concluded in Section 5.

This paper presents a continuous-time stochastic asset pricing model of a speculative financial market with fundamentalists and chartists. By applying stochastic bifurcation theory, we examine the limiting market equilibrium distribution numerically. We have shown that speculative behaviour of chartists can lead the market price to display different forms of equilibrium distributions. We have demonstrated the combined analysis of both D- and P-bifurcations certainly gives us a relatively complete picture of the stochastic behaviour of our model. In particular, when the chartists extrapolate the trend weakly (so that c<cDc<cD), the system only has one invariant measure View the MathML sourceδx∗(ω) which is stable. In this case, View the MathML sourcex∗(ω) has a stationary measure which has one peak. However, when the chartists extrapolate the trend strongly (so that c>cDc>cD), a new stable random Dirac measure View the MathML sourceδx♯(ω) appears and the corresponding stationary measure has a crater-like density. The change of stationary distribution is characterized by a bimodal logarithm price distribution and fat tails. To conclude the paper, we refer to Chiarella, He and Zheng [6] for more detailed analysis of the stochastic bifurcation of the fundamentalist-chartist model and the relationship between the stochastic and deterministic dynamics of the model. The cited paper provides more insights into the model of stochastic behaviour through stochastic bifurcation analysis, stochastic approximation methods and the difference between deterministic and stochastic dynamics. It demonstrates that stochastic bifurcation analysis can be a powerful tool to help in understanding various types of financial market behaviour.