Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence of an equilibrium with limited stock market participation and power utilities

Paolo Guasoni, Kasper Larsen, Giovanni Leoni

Abstract

For constants $γ\in (0,1)$ and $A\in (1,\infty)$, we prove existence and uniqueness of a solution to the singular and path-dependent Riccati-type ODE \begin{align*} \begin{cases} h'(y) = \frac{1+γ}{y}\big( γ- h(y)\big)+h(y)\frac{γ+ \big((A-γ)e^{\int_y^1 \frac{1-h(q)}{1-q}dq}-A\big)h(y)}{1-y},\quad y\in(0,1), h(0) = γ, \quad h(1) = 1. \end{cases} \end{align*} As an application, we use the ODE solution to prove existence of a Radner equilibrium with homogenous power-utility investors in the limited participation model from Basak and Cuoco (1998).

Existence of an equilibrium with limited stock market participation and power utilities

Abstract

For constants and , we prove existence and uniqueness of a solution to the singular and path-dependent Riccati-type ODE \begin{align*} \begin{cases} h'(y) = \frac{1+γ}{y}\big( γ- h(y)\big)+h(y)\frac{γ+ \big((A-γ)e^{\int_y^1 \frac{1-h(q)}{1-q}dq}-A\big)h(y)}{1-y},\quad y\in(0,1), h(0) = γ, \quad h(1) = 1. \end{cases} \end{align*} As an application, we use the ODE solution to prove existence of a Radner equilibrium with homogenous power-utility investors in the limited participation model from Basak and Cuoco (1998).
Paper Structure (21 sections, 18 theorems, 277 equations, 1 figure)

This paper contains 21 sections, 18 theorems, 277 equations, 1 figure.

Table of Contents

  1. Introduction
  2. ODE existence
  3. Auxiliary ODE results
  4. Local ODE existence for all $\xi$
  5. Global ODE existence for $\xi$ small
  6. Explosion for $\xi$ big
  7. Lipschitz estimates
  8. Global ODE existence for critical $\xi$
  9. Application to incomplete Radner equilibrium theory
  10. Autonomous state process $Y$
  11. Left-boundary point for $Y$
  12. Right-boundary point for $Y$
  13. Proof of strong SDE existence of $Y$
  14. Investors' maximization problems
  15. Radner equilibrium
  16. ...and 6 more sections

Key Result

Theorem 1.1

For $\gamma \in (0,1)$ and $A\in (1,\infty)$, there exists a unique solution $h \in {\mathcal{C}}^1([0,1])$ of KEY_ODE satisfying $\gamma \le h \le 1$.

Figures (1)

  • Figure 1: ODE solutions $h_{\xi_n}$ for $\xi_n \uparrow \sup \Xi$. The parameters are $\gamma:=0.5, \sigma_D:=0.2, A:=2$, and $\xi_n \in \{0.15, 0.152, 0.1522, 0.15223, 0.152232\}$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Lemma 2.1
  • Theorem 2.2
  • Remark 2.1
  • Theorem 2.3: Comparison
  • Theorem 2.4
  • Remark 2.2
  • Theorem 2.5
  • Theorem 2.6
  • Lemma 2.7
  • ...and 12 more