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Equilibrium strategies in time-inconsistent stochastic control problems with constraints: necessary conditions

Elisa Mastrogiacomo, Marco Tarsia

Abstract

This paper is concerned with a time-inconsistent recursive stochastic control problems where the forward state process is constrained through an additional recursive utility system. By adapting the Ekeland variational principle, necessary conditions for equilibrium strategies are presented concerning a second-order Hamiltonian function defined by pairs of backward stochastic differential equations. At last, we consider a finite horizon state constrained investment-consumption problem with non-exponential actualisation as an example to show the application in finance. The class of constraints investigated here includes the possibility of imposing a risk bound on the terminal value of the wealth process.

Equilibrium strategies in time-inconsistent stochastic control problems with constraints: necessary conditions

Abstract

This paper is concerned with a time-inconsistent recursive stochastic control problems where the forward state process is constrained through an additional recursive utility system. By adapting the Ekeland variational principle, necessary conditions for equilibrium strategies are presented concerning a second-order Hamiltonian function defined by pairs of backward stochastic differential equations. At last, we consider a finite horizon state constrained investment-consumption problem with non-exponential actualisation as an example to show the application in finance. The class of constraints investigated here includes the possibility of imposing a risk bound on the terminal value of the wealth process.

Paper Structure

This paper contains 4 sections, 5 theorems, 117 equations.

Table of Contents

  1. Problem Formulation
  2. Constrained Problem: Necessary Conditions
  3. Application to Portfolio Management
  4. Conclusions

Key Result

Theorem 1

Let us suppose that $U$ is bounded. Let ${\mathopen{(}}\bar{\bm{u}}(\:\!\bm{\cdot}\:\!),\bar{x}(\:\!\bm{\cdot}\:\!),\bar{y}(\:\!\bm{\cdot}\,;t),\bar{z}(\:\!\bm{\cdot}\,;t){\mathclose{)}}$ be an equilibrium 4-tuple such that the pair ${\mathopen{(}}\bar{\bm{u}}(\:\!\bm{\cdot}\:\!),\bar{x}(\:\!\bm{\cd Then, there exist two multipliers$\psi \equiv \psi_{t,a},\bm{\psi} \equiv \bm{\psi}_{t,a} \in \mat

Theorems & Definitions (17)

  • Definition 1: State constraint
  • Remark 1
  • Theorem 1: Maximum principle
  • Remark 2
  • Definition 2: Metric $\mathop{\mathrm{\mathbf{d}}}\nolimits$ on $\mathscr{U}\lbrack t,T\rbrack$
  • Remark 3
  • Lemma 1
  • Remark 4
  • Lemma 2
  • Remark 5
  • ...and 7 more