Slow-fast analysis of a multi-group asset flow model with implications for the dynamics of wealth

The multi-group asset flow model is a nonlinear dynamical system originally developed as a tool for understanding the behavioral foundations of market phenomena such as flash crashes and price bubbles. In this paper we use a modification of this model to analyze the dynamics of a single-asset market in situations when the trading rates of investors (i.e., their desire to exchange stock for cash) are prescribed ahead of time and independent of the state of the market. Under the assumption of fast trading compared to the time-rate of change in the prescribed trading rates we decompose the dynamics of the system to fast and slow components. We use the model to derive a variety of observations regarding the dynamics of price and investors' wealth, and the dependence of these quantities on the prescribed trading rates. In particular, we show that strategies with constant trading rates, which represent the well-known constant-rebalanced portfolio (CRP) strategies, are optimal in the sense that they minimize investment risks. In contrast, we show that investors pursuing non-CRP strategies are at risk of loss of wealth, as a result of the slow system not being integrable in the sense that cyclic trading rates do not always result in periodic price variations.