[Paper Review] Representation of homothetic forward performance processes via ergodic and infinite horizon quadratic BSDE in stochastic factor models

Gechun Liang, Thaleia Zariphopoulou|arXiv (Cornell University)|Nov 1, 2015

Stochastic processes and financial applications40 references3 citations

TL;DR

This paper derives representations of homothetic forward performance processes in incomplete stochastic factor models using ergodic and infinite-horizon quadratic backward stochastic differential equations (BSDEs). It establishes connections to risk-sensitive control and asymptotic value functions, showing convergence to traditional value functions over large horizons.

ABSTRACT

In an incomplete market, with incompleteness stemming from stochastic factors imperfectly correlated with the underlying stocks, we derive representations of homothetic forward investment performance processes (power, exponential and logarithmic). We develop a connection with ergodic and infinite horizon quadratic BSDE, and with a risk-sensitive control problem. We also develop a connection, for large trading horizons, with a family of traditional homothetic value function processes.

Motivation & Objective

• To characterize homothetic forward performance processes in incomplete markets driven by stochastic factors correlated imperfectly with risky assets.
• To establish a connection between forward performance processes and solutions of ergodic and infinite-horizon quadratic BSDEs.
• To link the forward performance framework to risk-sensitive stochastic control problems.
• To analyze the asymptotic behavior of forward performance processes as the trading horizon tends to infinity.
• To show convergence of forward performance processes to classical homothetic value function processes in the long-horizon limit.

Proposed method

• Formulate the forward performance process as a solution to a quadratic BSDE with random terminal condition in a stochastic factor model.
• Utilize the theory of ergodic BSDEs to characterize the long-term behavior of the performance process under suitable ergodicity assumptions on the factor process.
• Establish existence and uniqueness of solutions to infinite-horizon quadratic BSDEs under appropriate integrability and growth conditions.
• Connect the forward performance process to a risk-sensitive control problem via the dynamic programming principle and verification arguments.
• Derive asymptotic expansions of the forward performance process as the time horizon diverges, showing convergence to traditional value functions.
• Employ homothetic structure to reduce the infinite-dimensional control problem to a finite-dimensional one via scaling properties.

Experimental results

Research questions

• RQ1 How can homothetic forward performance processes be represented in incomplete stochastic factor models with correlated factors?
• RQ2 What is the connection between forward performance processes and solutions of ergodic and infinite-horizon quadratic BSDEs?
• RQ3 How do forward performance processes relate to risk-sensitive control problems in this setting?
• RQ4 What is the asymptotic behavior of forward performance processes as the trading horizon tends to infinity?
• RQ5 Do forward performance processes converge to classical homothetic value function processes in the long-horizon limit?

Key findings

• Homothetic forward performance processes in stochastic factor models can be represented via solutions to infinite-horizon quadratic BSDEs under suitable conditions on the factor process.
• The existence and uniqueness of solutions to the relevant ergodic and infinite-horizon quadratic BSDEs are established under appropriate integrability and non-degeneracy assumptions.
• A deep connection is established between the forward performance process and a risk-sensitive control problem, with the value function of the latter being linked to the solution of the BSDE.
• As the trading horizon diverges, the forward performance process converges to a classical homothetic value function process, confirming consistency with traditional dynamic programming frameworks.
• The homothetic structure enables a reduction of the forward performance problem to a finite-dimensional system, facilitating explicit characterization in terms of the underlying factor dynamics.
• The asymptotic convergence is shown to hold under mild regularity and ergodicity conditions on the stochastic factors, extending classical results to the forward performance setting.