Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on time-inconsistent stochastic control problems with higher-order moments

Yike Wang

TL;DR

The paper tackles time-inconsistent stochastic control with higher-order moments by formulating a mean-variance baseline plus a general deterministic function of higher moments, and derives both sufficiency and necessity of an open-loop Nash equilibrium control (ONEC) under relaxed regularity. It employs a perturbation argument, a flow of linear BSDEs that generates diagonal processes, and an extended stochastic Lebesgue differentiation theorem to handle non-square-integrable data, culminating in a characterization of non-trivial ONECs via an explicit equilibrium condition. A key result shows that, under state-diffusion independence (I≡0) and a homogeneity condition on the higher-moment function, the mean-variance time-consistent solution is itself an ONEC for the higher-moment problem; the analysis also delineates when this equivalence fails or reduces to simpler integral equations. The work expands the applicability of time-consistent control with higher-order risk measures and provides a roadmap for future exploration of existence, uniqueness, and more general dynamics, while preserving rigorous connections to BSDEs and stochastic differentiation tools.

Abstract

In this paper, we extend the research on time-consistent stochastic control problems with higher-order moments, as formulated by [Y. Wang et al. SIAM J. Control. Optim., 63 (2025), in press]. We consider a linear controlled dynamic equation with state-dependent diffusion, and let the sum of a conventional mean-variance utility and a fairly general function of higher-order central moments be the objective functional. We obtain both the sufficiency and necessity of the equilibrium condition for an open-loop Nash equilibrium control (ONEC), under some continuity and integrability assumptions that are more relaxed and natural than those employed before. Notably, we derive an extended version of the stochastic Lebesgue differentiation theorem for necessity, because the equilibrium condition is represented by some diagonal processes generated by a flow of backward stochastic differential equations whose the data do not necessarily satisfy the usual square-integrability. Based on the derived equilibrium condition, we obtain the algebra equation for a deterministic ONEC. In particular, we find that the mean-variance equilibrium strategy is an ONEC for our higher-order moment problem if and only if the objective functional satisfies a homogeneity condition.

A note on time-inconsistent stochastic control problems with higher-order moments

TL;DR

The paper tackles time-inconsistent stochastic control with higher-order moments by formulating a mean-variance baseline plus a general deterministic function of higher moments, and derives both sufficiency and necessity of an open-loop Nash equilibrium control (ONEC) under relaxed regularity. It employs a perturbation argument, a flow of linear BSDEs that generates diagonal processes, and an extended stochastic Lebesgue differentiation theorem to handle non-square-integrable data, culminating in a characterization of non-trivial ONECs via an explicit equilibrium condition. A key result shows that, under state-diffusion independence (I≡0) and a homogeneity condition on the higher-moment function, the mean-variance time-consistent solution is itself an ONEC for the higher-moment problem; the analysis also delineates when this equivalence fails or reduces to simpler integral equations. The work expands the applicability of time-consistent control with higher-order risk measures and provides a roadmap for future exploration of existence, uniqueness, and more general dynamics, while preserving rigorous connections to BSDEs and stochastic differentiation tools.

Abstract

In this paper, we extend the research on time-consistent stochastic control problems with higher-order moments, as formulated by [Y. Wang et al. SIAM J. Control. Optim., 63 (2025), in press]. We consider a linear controlled dynamic equation with state-dependent diffusion, and let the sum of a conventional mean-variance utility and a fairly general function of higher-order central moments be the objective functional. We obtain both the sufficiency and necessity of the equilibrium condition for an open-loop Nash equilibrium control (ONEC), under some continuity and integrability assumptions that are more relaxed and natural than those employed before. Notably, we derive an extended version of the stochastic Lebesgue differentiation theorem for necessity, because the equilibrium condition is represented by some diagonal processes generated by a flow of backward stochastic differential equations whose the data do not necessarily satisfy the usual square-integrability. Based on the derived equilibrium condition, we obtain the algebra equation for a deterministic ONEC. In particular, we find that the mean-variance equilibrium strategy is an ONEC for our higher-order moment problem if and only if the objective functional satisfies a homogeneity condition.

Paper Structure

This paper contains 9 sections, 8 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Mathematical preliminaries
  4. Perturbation argument
  5. BSDEs and diagonal processes
  6. Stochastic Lebesgue differentiation theorem
  7. Sufficient and necessary conditions for non-trivial ONECs
  8. Closed-form solution
  9. Concluding remarks

Key Result

Lemma 3.1

For any given non-trivial $\bar{u} \in \mathbb{L}_{\mathbb{F}}^{2} ( 0,T; \mathbb{L}^{2n} ( \Omega ) )$, $t \in [ 0,T )$, $\zeta \in \mathbb{L}^{2n}_{ \mathcal{F}_{t} } ( \Omega )$ and a sufficiently small $\varepsilon > 0$,

Theorems & Definitions (11)

  • Definition 2.1
  • Lemma 3.1
  • Lemma 3.3
  • Lemma 3.5
  • Theorem 4.1
  • Remark 4.2
  • Theorem 4.3
  • Lemma 5.1
  • Theorem 5.2
  • Theorem 5.3
  • ...and 1 more