[Paper Review] Systems of ergodic BSDEs arising in regime switching forward performance processes

Ying Hu, Gechun Liang+1 more|arXiv (Cornell University)|Jul 4, 2018

Stochastic processes and financial applications32 references20 citations

TL;DR

This paper introduces and solves a new class of multidimensional backward stochastic differential equations (BSDEs) with ergodic behavior, termed ergodic BSDE systems, to characterize forward performance processes and optimal strategies in regime-switching financial markets. The key contribution is proving the existence and uniqueness of solutions under quadratic growth and regime-switching dynamics, linking them to long-term growth rates in utility maximization and the large-time behavior of PDEs with quadratic Hamiltonians.

ABSTRACT

We introduce and solve a new type of quadratic backward stochastic differential equation systems defined in an infinite time horizon, called \\emph{ergodic BSDE systems}. Such systems arise naturally as candidate solutions to characterize forward performance processes and their associated optimal trading strategies in a regime switching market. In addition, we develop a connection between the solution of the ergodic BSDE system and the long-term growth rate of classical utility maximization problems, and use the ergodic BSDE system to study the large time behavior of PDE systems with quadratic growth Hamiltonians.

Motivation & Objective

• To develop a new class of infinite-horizon BSDE systems that model forward performance processes in financial markets with multiple regimes.
• To establish the existence and uniqueness of solutions to these ergodic BSDE systems under quadratic growth and regime-switching dynamics.
• To connect the solutions of the ergodic BSDE systems to the long-term growth rates in classical utility maximization problems.
• To analyze the large-time behavior of PDE systems with quadratic Hamiltonians using the ergodic BSDE framework.
• To extend the theory of Markovian forward performance processes to include regime-switching environments via coupled BSDE systems.

Proposed method

• Formulate a system of infinite-horizon quadratic BSDEs with regime-switching dynamics, where the driver includes interaction terms between regimes through transition rates.
• Use a comparison theorem for multidimensional BSDEs to establish uniqueness and stability of solutions under boundedness and growth conditions.
• Apply a change of measure technique via stochastic exponentials to transform the original probability space and simplify the dynamics under an equivalent measure.
• Leverage moment and coupling estimates to control the growth of solutions and prove convergence to zero over infinite time horizons.
• Construct a Markovian representation of the solution via measurable functions of the underlying diffusion and Markov chain processes.
• Utilize the ergodic limit of the value function to identify the long-run growth rate, linking the solution to ergodic control problems.

Experimental results

Research questions

• RQ1 Can a system of infinite-horizon BSDEs be formulated to model forward performance processes in a regime-switching market?
• RQ2 What conditions ensure the existence and uniqueness of solutions to such ergodic BSDE systems with quadratic growth and regime interactions?
• RQ3 How do the solutions of these ergodic BSDE systems relate to the long-term growth rate in utility maximization problems?
• RQ4 What is the large-time behavior of PDE systems with quadratic Hamiltonians, and how can it be characterized via ergodic BSDEs?
• RQ5 Can the Markovian representation of forward performance processes be extended to include regime-switching dynamics through coupled BSDE systems?

Key findings

• The ergodic BSDE system admits a unique bounded solution under appropriate conditions on the driver, including quadratic growth in the control variable and regime-switching interaction terms.
• The solution to the ergodic BSDE system characterizes the long-term growth rate of the value function in classical utility maximization problems, linking stochastic control to forward performance processes.
• The solution components exhibit at most linear growth, ensuring stability and convergence over infinite time horizons.
• The difference between two solutions corresponding to different controls converges to zero as time goes to infinity, implying uniqueness in the ergodic limit.
• Moment and coupling estimates show that the solution decays exponentially in time when perturbed, confirming stability and robustness under perturbations of the initial state.
• The framework enables the construction of Markovian forward performance processes in regime-switching markets via the solution of the ergodic BSDE system.