Local stability analysis of a stochastic evolutionary financial market model with a risk-free asset

Abstract

This paper introduces and analyzes an evolutionary model of a financial marketwith a risk-free asset. Focus is on the study of local stability of the wealth dynamics through the application of recent results on the linearization and stability of random dynamical systems (Evstigneev et al. Proc Am Math Soc 139:1061-1072. Conditions are derived for the linearization of the model at an equilibrium state which ensure local convergence of sample paths to this equilibrium. The paper also shows that the concept of local stability is closely related to the notion of evolutionary stability. A locally evolutionarily stable investmentstrategy in the evolutionary model with a risk-free asset is derived, extending previousresearch. The method illustrated here is applicable for the analysis of manifold economic and financial dynamic models involving randomness. Conditions are derived for the linearization of the model at an equilibrium state which ensure local convergence of sample paths to this equilibrium. The paper also shows that concept of local stability is closely related to the notion of evolutionary stability. A locally evolutionarily stable investment strategy in the evolutionary model with a risk-free asset is derived, extending previous research. Our analysis can serve as an illustration of a mathematically correct approach to the study of local stability in stochastic systems. The method will be useful in the study of other economic and financial dynamic models involving randomness.JEL classification: G11, G12.

Asset Prices and Wealth Dynamics in a Financial Market with Endogenous Liquidation Risk

“…The literature has emerged with the aim of modelling possible sources of endogenous risk. In particular, models close to ours that investigate the joint asset prices and wealth dynamics are Chiarella and He (2001), Chiarella et al (2006), Anufriev et al (2006), Anufriev and Bottazzi (2010), Anufriev and Dindo (2010), Evstigneev et al (2011), and Anufriev et al (2012). However, to the best of our knowledge, no contribution has so far analysed the transformation of random liquidation needs in liquidation risk and its impact on the asset price dynamics.…”

“…process; this directly implies that any change in the price of the risky security causes an essentially instantaneous adjustment in the paid dividend of an identical magnitude. Evstigneev et al (2011) instead anchor the dividend to the aggregate wealth in the economy. As opposed to linking the dividend to the endogenous dynamics of the economy, Chiarella et al (2006) and Anufriev and Dindo (2010) implement an exogenously growing dividend process with i.i.d.…”

An evolutionary finance model with a risk-free asset

“…Furthermore, only local evolutionary stability results, rather than global ones, are obtained, so that the fundamental goals typically considered in evolutionary finance are achieved only partially. Finally, the model proposed in [21] required specific new techniques for its analysis, while here we derive our results from those obtained in Amir et al [3] in the context of purely risky assets, which simplifies proofs substantially.…”

“…An earlier attempt to build an evolutionary finance model with a risk-free asset was undertaken in [21]. There are substantial distinctions between that study and the present one, among which the most notable ones are as follows.…”

“…The model in [21] describes a market with long-lived (rather than short-lived) dividend-paying assets. Although it is also assumed that the dividends are scalable, they are supposed to depend linearly on the total market wealth, rather than total cash supply.…”