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Entropy-Maximizing Dynamics of Continuous Markets

Eckhard Platen

Abstract

By assuming the existence of the growth optimal portfolio (GOP), the stationarity of GOP-volatilities, and the maximization of relative entropy, the paper applies the benchmark approach to the modeling of the long-term dynamics of continuous markets. It reveals conservation laws, where the GOP is shown to follow a time-transformed squared Bessel process of dimension four. Moreover, it predicts the convergence of the averages of the GOP-volatilities with respect to the driving independent Brownian motions toward a common level.

Entropy-Maximizing Dynamics of Continuous Markets

Abstract

By assuming the existence of the growth optimal portfolio (GOP), the stationarity of GOP-volatilities, and the maximization of relative entropy, the paper applies the benchmark approach to the modeling of the long-term dynamics of continuous markets. It reveals conservation laws, where the GOP is shown to follow a time-transformed squared Bessel process of dimension four. Moreover, it predicts the convergence of the averages of the GOP-volatilities with respect to the driving independent Brownian motions toward a common level.
Paper Structure (5 sections, 3 theorems, 72 equations)

This paper contains 5 sections, 3 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Basis Market
  3. Entropy Maximization
  4. Activity Equilibrium
  5. Conclusion

Key Result

Theorem 3.2

For a basis market with maximized entropy ${\cal{H}}(\bar{q}^j_\infty)$ the function $\phi^j(.)$ takes the form the respective stationary density is the gamma density with conserved four degrees of freedom, conserved logarithmic average and conserved arithmetic average and the $j$-th normalized GOP satisfies the SDE for $t \in[0,\infty)$, with random initial value $Y^j_{\tau^j_0}$ distributed

Theorems & Definitions (6)

  • Definition 2.1
  • Definition 3.1
  • Theorem 3.2: Normalized GOP Dynamics
  • Theorem 4.1: Activity Theorem
  • Definition 4.2
  • Theorem 4.3