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On stochastic control problems with higher-order moments

Yike Wang, Jingzhen Liu, Alain Bensoussan, Ka-Fai Cedric Yiu, Jiaqin Wei

TL;DR

This work studies time-inconsistent stochastic control where the objective depends on the mean and higher-order central moments of the terminal state. It develops two complementary time-consistent solution concepts: closed-loop Nash equilibrium controls (CNEC) via a PDE system that obviates the equilibrium value function, and open-loop Nash equilibrium controls (ONEC) via a maximum principle leading to forward–backward SDEs. In the linear-controlled-SDE setting, both CNEC and ONEC become state- and path-independent and can be expressed in a simple affine form, with an auxiliary moment-equation determining the feedback gain; several limiting cases yield explicit closed-form controls and equilibrium values. The analysis provides a versatile framework for distributional preferences, including mean–variance–type and even/ambiguous penalty structures, with practical closed-form solutions in several scenarios and a clear pathway to numerical implementation in more general settings.

Abstract

In this paper, we focus on a class of time-inconsistent stochastic control problems, where the objective function includes the mean and several higher-order central moments of the terminal value of state. To tackle the time-inconsistency, we seek both the closed-loop and the open-loop Nash equilibrium controls as time-consistent solutions. We establish a partial differential equation (PDE) system for deriving a closed-loop Nash equilibrium control, which does not include the equilibrium value function and is different from the extended Hamilton-Jacobi-Bellman (HJB) equations as in Björk et al. (Finance Stoch. 21: 331-360, 2017). We show that our PDE system is equivalent to the extended HJB equations that seems difficult to be solved for our higher-order moment problems. In deriving an open-loop Nash equilibrium control, due to the non-separable higher-order moments in the objective function, we make some moment estimates in addition to the standard perturbation argument for developing a maximum principle. Then, the problem is reduced to solving a flow of forward-backward stochastic differential equations. In particular, we investigate linear controlled dynamics and some objective functions affine in the mean. The closed-loop and the open-loop Nash equilibrium controls are identical, which are independent of the state value, random path and the preference on the odd-order central moments. By sending the highest order of central moments to infinity, we obtain the time-consistent solutions to some control problems whose objective functions include some penalty functions for deviation.

On stochastic control problems with higher-order moments

TL;DR

This work studies time-inconsistent stochastic control where the objective depends on the mean and higher-order central moments of the terminal state. It develops two complementary time-consistent solution concepts: closed-loop Nash equilibrium controls (CNEC) via a PDE system that obviates the equilibrium value function, and open-loop Nash equilibrium controls (ONEC) via a maximum principle leading to forward–backward SDEs. In the linear-controlled-SDE setting, both CNEC and ONEC become state- and path-independent and can be expressed in a simple affine form, with an auxiliary moment-equation determining the feedback gain; several limiting cases yield explicit closed-form controls and equilibrium values. The analysis provides a versatile framework for distributional preferences, including mean–variance–type and even/ambiguous penalty structures, with practical closed-form solutions in several scenarios and a clear pathway to numerical implementation in more general settings.

Abstract

In this paper, we focus on a class of time-inconsistent stochastic control problems, where the objective function includes the mean and several higher-order central moments of the terminal value of state. To tackle the time-inconsistency, we seek both the closed-loop and the open-loop Nash equilibrium controls as time-consistent solutions. We establish a partial differential equation (PDE) system for deriving a closed-loop Nash equilibrium control, which does not include the equilibrium value function and is different from the extended Hamilton-Jacobi-Bellman (HJB) equations as in Björk et al. (Finance Stoch. 21: 331-360, 2017). We show that our PDE system is equivalent to the extended HJB equations that seems difficult to be solved for our higher-order moment problems. In deriving an open-loop Nash equilibrium control, due to the non-separable higher-order moments in the objective function, we make some moment estimates in addition to the standard perturbation argument for developing a maximum principle. Then, the problem is reduced to solving a flow of forward-backward stochastic differential equations. In particular, we investigate linear controlled dynamics and some objective functions affine in the mean. The closed-loop and the open-loop Nash equilibrium controls are identical, which are independent of the state value, random path and the preference on the odd-order central moments. By sending the highest order of central moments to infinity, we obtain the time-consistent solutions to some control problems whose objective functions include some penalty functions for deviation.

Paper Structure

This paper contains 13 sections, 12 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Model and problem formulation
  3. Sufficient condition for CNEC: extended HJB-PDE
  4. Sufficient condition for ONEC: maximum principle
  5. Particular cases with linear controlled SDE
  6. Closed-loop Nash equilibrium control
  7. Open-loop Nash equilibrium control
  8. Limiting cases
  9. Limiting case I: mean-exponential utility function
  10. Limiting case II: hyperbolic cosine penalty function
  11. Limiting case III: cosine penalty function
  12. Limiting case IV: ambiguous cosine penalty function
  13. Concluding remark

Key Result

Lemma 3.1

Fix $\tilde{u} \in \mathcal{U}$ and $( s,y, \vec{z} ) \in [ 0,T ) \times \mathbb{R} \times \mathbb{R}^{n}$. Assume that ${U}^{ s,y, \vec{z} }$ and $\vec{m} := ( {m}^{(1)}, \ldots, {m}^{(n)} )$ are the classical solutions of the following PDEs on $[ 0,T ] \times \mathbb{R}$: respectively; and for any $( t,x ) \in [ 0,T ) \times \mathbb{R}$, Then, ${U}^{ s,y, \vec{z} } ( t,x ) = \mathbb{E}_{t} [ \

Theorems & Definitions (25)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3: Verification theorem
  • Lemma 4.1
  • ...and 15 more