Stationary Distributions of the Mode-switching Chiarella Model

TL;DR

This work derives the stationary distributions of the extended Chiarella model across multiple dynamical regimes, focusing on the mispricing $p(\delta)$ and trend $p(M)$ distributions under competing mean-reversion and trend-following forces. Using Fokker–Planck analysis, Lyapunov equations, and Furutsu–Novikov methods, the authors show that in the linear/ small-noise limit both $p(\delta)$ and $p(M)$ are Gaussian, while in slow-trend regimes the mispricing attains a Gaussian–cosh form with a P-bifurcation controlled by the balance between mean reversion and trend strength and noise level. In fast-trend limits ($\alpha\gg\kappa$) the analysis yields unimodal Gaussian mispricing for weak coupling (via an effective telegraphic-noise description) and reveals that bimodality in the trend distribution does not generically imply bimodality in mispricing when noise is present; only at sufficiently strong feedback do both distributions become bimodal. The results contest previous claims linking P-bifurcation to Hopf bifurcation of the noiseless system and provide a more nuanced criterion that depends on $\kappa$, $\beta$, and the composite noise $\sigma^2$. Together, these findings clarify when stationary distributions are uni- or multimodal and illuminate how market-like regimes of value investors and trend followers shape mispricing dynamics.

Abstract

We derive the stationary distribution in various regimes of the extended Chiarella model of financial markets. This model is a stochastic nonlinear dynamical system that encompasses dynamical competition between a (saturating) trending and a mean-reverting component. We find the so-called mispricing distribution and the trend distribution to be unimodal Gaussians in the small noise, small feedback limit. Slow trends yield Gaussian-cosh mispricing distributions that allow for a P-bifurcation: unimodality occurs when mean-reversion is fast, bimodality when it is slow. The critical point of this bifurcation is established and refutes previous ad-hoc reports and differs from the bifurcation condition of the dynamical system itself. For fast, weakly coupled trends, deploying the Furutsu-Novikov theorem reveals that the result is again unimodal Gaussian. For the same case with higher coupling we disprove another claim from the literature: bimodal trend distributions do not generally imply bimodal mispricing distributions. The latter becomes bimodal only for stronger trend feedback. The exact solution in this last regime remains unfortunately beyond our proficiency.

Figures (4)

• Figure 1: Grey: Numerical histograms of the stationary distribution in the case $\gamma$ small. Simulation parameters are $(\kappa,\,\beta,\,\alpha,\,\sigma_N,\,\sigma_V)=(0.1,\,0.2,\,0.2,\,0.2,\,0.1)$ and $\gamma$ as detailed in the plot. $T=\gamma\times 10^9$, d$t=\gamma / 2$ and $g=0$. Coloured: Corresponding analytical stationary distributions $p(\delta)$ according to Eq.\ref{['eq: statdist_betagamma_small']}. The distributions with $\gamma >10^{-4}$ are shifted by multiples of 1 on the abscissa and of 0.5 on the ordinate.
• Figure 2: Same as Fig. \ref{['fig:stat_dist_betagammasmall']} but for $\alpha\ll\kappa$, while $\gamma$ large, with $p(\delta)$ according to Eq.\ref{['eq:stattdist_largegamma']}. Numerical parameters are $(\alpha,\,\beta,\,\gamma,\,\sigma_N,\,\sigma_V)=(2\times 10^{-5},\, 0.05,\, 5\times 10^{4},\,0.2,\,0.1)$ and $\kappa$ detailed in the plot. $T=5\times 10^7$, d$t=0.01$.
• Figure 3: Same as Fig. \ref{['fig:stat_dist_betagammasmall']} but in the case $\alpha\gg\kappa$ with $p(\delta)$ according to Eq.\ref{['eq:stat_dist_alpha_large']}. Numerical parameters are $(\alpha,\,\gamma,\,\sigma_N,\,\sigma_V)=(500,\,1,\,0.8,\,0.1)$, while $\kappa$ and $\beta$ are detailed in the plot. $T=10^5$, d$t=10^{-3}$.
• Figure 4: Same as Fig. \ref{['fig:stat_dist_betagammasmall']} but for $\alpha,\,\beta\gg\kappa$, and for both $p(M)$ (left) and $p(\delta)$ (right). Top: Numerical parameters are $(\alpha,\,\kappa,\,\gamma,\,\sigma_N,\,\sigma_V)=(50,\, 0.05, 1,\,0.7,\,0.2)$, $\beta = 5$ and drift $g=0$, corresponding to $\Theta \approx 1.01$ and $\gamma \sigma_N \sqrt{\alpha} \approx 5$. $T=10^5$, d$t=0.001$. We clearly see that for this set of parameters $p(M)$ is bimodal while $p(\delta)$ is still unimodal. Bottom: same parameters as before except $\beta = 18$, such that $\Theta \approx 3.64$, in which case both distributions are bimodal. We have no exact analytical predictions to compare with in these cases.