[Paper Review] Asymptotic distribution of capital in a model of an investment market with competition

Mikhail Zhitlukhin|arXiv (Cornell University)|Jan 1, 2018

Stochastic processes and financial applicationsEconomics, Econometrics and Finance11 references3 citations

TL;DR

This paper studies a continuous-time stochastic game-theoretic model of an investment market with endogenous asset pricing, where investors compete for dividend income. It proves the existence of a dominant strategy that ensures an investor maintains a positive capital share indefinitely, dominates relative asset prices, and accumulates wealth faster than competitors regardless of their strategies.

ABSTRACT

We consider a stochastic game-theoretic model of an investment market in continuous time where investors compete for dividend income from several assets. Asset prices are determined endogenously from the equality of supply and demand. The main results are related to the question in what proportions the total capital will be distributed among the investors, and what will be the prices of the assets asymptotically on the infinite time horizon depending on the strategies of the investors. We prove that there exists a strategy with the following properties: the proportion of its capital on the entire time horizon is separated from zero with probability 1 regardless of the strategies of competitors; the relative asset prices are asymptotically determined by it; the proportion of capital of investors who follow essentially different strategies tends to zero. We also show that investors who follow this strategy will accumulate capital faster than their competitors under several definitions of the speed of capital growth.

Motivation & Objective

• To analyze the asymptotic distribution of capital among competing investors in a continuous-time stochastic investment market.
• To determine how endogenously determined asset prices evolve over an infinite time horizon under different investor strategies.
• To identify a strategy that guarantees a non-vanishing share of total capital regardless of competitors’ actions.
• To investigate conditions under which certain strategies outperform others in terms of capital growth speed.

Proposed method

• Modeling the investment market as a stochastic game in continuous time with random dividend income from multiple assets.
• Formulating endogenous asset pricing through market clearing, where supply equals demand for each asset.
• Analyzing the long-run behavior of capital proportions using probabilistic methods and stochastic control techniques.
• Defining a specific strategy that maintains a positive lower bound on capital share almost surely over time.
• Comparing growth rates of capital shares under different strategies using multiple definitions of wealth accumulation speed.
• Applying martingale and large deviation arguments to prove asymptotic dominance of the proposed strategy.

Experimental results

Research questions

• RQ1 What proportion of total capital will be held by an investor following a given strategy as time approaches infinity?
• RQ2 How do endogenously determined asset prices behave asymptotically under competing investor strategies?
• RQ3 Is there a strategy that ensures a positive lower bound on capital share regardless of competitors’ actions?
• RQ4 Under what conditions does one investor’s capital grow faster than others in terms of multiple growth speed definitions?
• RQ5 How do relative asset prices converge asymptotically in relation to the dominant investor’s strategy?

Key findings

• There exists a strategy such that the investor’s capital share remains bounded away from zero with probability 1 over the infinite time horizon.
• The relative prices of assets converge asymptotically to values determined solely by the dominant investor’s strategy.
• Investors following strategies that are essentially different from the dominant one see their capital shares converge to zero almost surely.
• The dominant strategy leads to faster capital accumulation than competitors under multiple formal definitions of growth speed.
• The asymptotic behavior of capital distribution and asset prices is fully determined by the strategy of the dominant investor.