Existence and uniqueness for Kryger et al.’s ODE system in the endogenous-habit scaled mean–variance model

Establish the existence and uniqueness of solutions to the ordinary differential equation system (21)–(25) proposed by Kryger, Steffensen, and Westergaard (2020) that characterizes intra-personal equilibrium strategies for portfolio selection under endogenous habit formation with scaled mean–variance preferences. Specifically, prove that this ODE system admits a unique solution on the full time horizon under the model’s assumptions, thereby validating the equilibrium characterization in that setting.

Background

The paper surveys several time-inconsistent preferences where equilibria are characterized by integral or differential equations. For the scaled mean–variance preferences with endogenous habit formation studied by Kryger et al. (2020), the equilibrium strategy is described by an ODE system (their equations (21)–(25)).

The authors explicitly note that, in the endogenous habit case, Kryger et al. did not prove existence and uniqueness of solutions to this ODE system, leaving the equilibrium characterization unvalidated from a well-posedness standpoint. The current paper provides a unified framework for integral-equation characterizations under minimal assumptions, but this specific ODE-based characterization from the literature remains unresolved.

Clarifying the existence and uniqueness of solutions to Kryger et al.’s ODE system would close a gap in the theoretical foundations of equilibrium strategies for endogenous habit, scaled mean–variance preferences and ensure the viability of their continuous-time equilibrium analysis.