Risk Sensitive Portfolio Optimization with Default Contagion and Regime-Switching

TL;DR

This paper addresses risk-sensitive portfolio optimization in a complex regime-switching credit market with default contagion, establishing existence, uniqueness, and approximation of optimal strategies via nonlinear dynamical programming equations.

Contribution

It introduces a novel approach to solve infinite-dimensional DPEs in regime-switching markets with default contagion, including a truncation method and convergence analysis for optimal strategies.

Findings

Proved existence and uniqueness of solutions to recursive DPEs.

Developed a truncation-based approximation method for infinite state spaces.

Established convergence of approximate value functions to the true solution.

Abstract

We study an open problem of risk-sensitive portfolio allocation in a regime-switching credit market with default contagion. The state space of the Markovian regime-switching process is assumed to be a countably infinite set. To characterize the value function, we investigate the corresponding recursive infinite-dimensional nonlinear dynamical programming equations (DPEs) based on default states. We propose to work in the following procedure: Applying the theory of monotone dynamical system, we first establish the existence and uniqueness of classical solutions to the recursive DPEs by a truncation argument in the finite state space. The associated optimal feedback strategy is characterized by developing a rigorous verification theorem. Building upon results in the first stage, we construct a sequence of approximating risk sensitive control problems with finite states and prove that the resulting smooth value functions will converge to the classical solution of the original system of DPEs. The construction and approximation of the optimal feedback strategy for the original problem are also thoroughly discussed.