Non-Autonomous Time-Consistent Control

• Non-autonomous time-consistent problems are defined in stochastic control with explicit time-dependent parameters that ensure an optimal control policy remains valid over time.
• A decomposition method splits the problem into a finite-horizon time-inconsistent segment and an infinite-horizon tail with exponential discounting solved via classical dynamic programming.
• Equilibrium strategies in this framework reconcile recursive cost structures, utilizing Hamilton-Jacobi-Bellman equations and backward stochastic differential equations for robust solutions.

A non-autonomous time-consistent problem in stochastic optimal control refers to a dynamic decision problem in which system parameters or performance criteria depend explicitly on time, yet the optimal control policy maintains time consistency—meaning that a strategy devised at the initial time remains optimal at all subsequent times under a suitable information structure or after appropriate augmentation of the state. The theory emerges prominently when infinite-horizon stochastic control problems with initially time-inconsistent preferences or constraints are decomposed under regularity or rationality assumptions, leading to a well-posed non-autonomous, time-consistent "tail" problem that can be addressed by dynamic programming.

1. Problem Decomposition and Setting

Infinite-horizon stochastic optimal control problems with general (possibly non-exponential) discounting or cost structures are not typically time consistent: an optimal control planned at time $t_0$ may lose its optimality as time evolves. However, if it is assumed that decision-makers become rational, or the cost functional simplifies (e.g., discounting becomes exponential) after some finite time $T_0$, then the problem can be rigorously split into two segments:

• Finite-Horizon Time-Inconsistent Segment ($[t, t+T_0]$): The system operates under general, typically non-exponential, possibly non-stationary discounting or cost structures, causing time-inconsistency.
• Infinite-Horizon Non-Autonomous Time-Consistent Tail ($[t+T_0, \infty)$): Beyond a threshold, the system cost reduces to a classical (e.g., exponentially discounted) form, and the usual time-consistent dynamic programming approach is valid.

Formally, this is encapsulated by imposing for $s \ge t+T_0$:

$X(s - t) = e^{-\delta(s-t)}, \qquad \delta > 0,$

where $X(\cdot)$ is the effective discounting applied beyond $T_0$ [(Wei et al., 18 Sep 2025), (1.15)].

2. Mathematical Formulation of the Non-Autonomous Time-Consistent Problem

On $[t+T_0, \infty)$, the control problem is recast as a classical non-autonomous stochastic optimal control problem. The system evolves as a controlled diffusion: $$dX(s) = b(s, X(s), u(s))\, ds + \sum_{i=1}^d \sigma_i(s, X(s), u(s))\, dW_i(s), \quad X(t) = x,$$ where $b$ and $\sigma_i$ are continuous in $(s, x, u)$ and satisfy uniform Lipschitz and non-degeneracy assumptions [(H1)].

The performance criterion is of Bolza type: $$J^0(t, x; u(\cdot)) = \mathbb{E}\left[\int_t^\infty e^{-\delta(s-t)}\, g(s, X(s), u(s))\, ds\right],$$ with $g$ the running cost, typically bounded or $|g(s, x, u)| \leq p(s)$ for $p \in L^1$ [(H2)].

The value function and associated HJB equation are: $V^0(t, x) = \inf_{u(\cdot)} J^0(t, x; u(\cdot)),$

$$V^0_t(t, x) + H\bigl(t, x, u^*, V^0(t,x), V^0_x(t,x), V^0_{xx}(t,x)\bigr) = 0,$$

where

$$H(t, x, u, p, P) = p \cdot b(t, x, u) + \tfrac{1}{2} \mathrm{trace}[(\sigma\sigma^\top)(t,x,u) P] + g(t, x, u).$$

The optimal (feedback) law is

$$u^*(s) = v\bigl(s, X(s)\bigr) = \argmin_{u} H\bigl(s, X(s), u, V^0(s, X(s)), V^0_x(s, X(s)), V^0_{xx}(s, X(s))\bigr) \tag{2.16}$$

The system is non-autonomous due to explicit $s$-dependence in $b$, $\sigma$, and $g$, but time consistency is restored by the exponential discount.

3. Equilibrium Strategy and Matching at the Interface

The overall infinite-horizon problem is handled by constructing an equilibrium strategy:

• The "tail" (non-autonomous, time-consistent) control $u^*$ is derived on $[t+T_0, \infty)$ via the time-consistent HJB theory.
• On $[t, t+T_0]$, where time-inconsistency persists, an "equilibrium HJB" or similar local equilibrium methodology is invoked to produce a policy $\bar{u}$ yielding local time-consistency.

The essential mechanism is to ensure that the solution at $t+T_0$—denoted $V^0(t+T_0, \cdot)$—serves as the continuation value for the earlier interval: $$J(t, x; u(\cdot)) = \mathbb{E}\left[\int_t^{t+T_0} A(s-t)\, g(s, X(s), u(s))\, ds + e^{-\delta T_0} V^0(t+T_0, X(t+T_0))\right], \tag{1.23}$$ where $A(\cdot)$ describes the pre-$T_0$ discount. The equilibrium strategy on $[t, t+T_0]$ is constructed such that $\bar{u}$ is locally optimal and consistent with the tail control policy (Wei et al., 18 Sep 2025).

4. Recursive Cost Structures

In addition to Bolza-type (additive) costs, the methodology extends to recursive cost problems formulated via backward stochastic differential equations (BSDEs). Consider

$dY(s) = -g\bigl(t, s, X(s), u(s), Y(s), Z(s)\bigr)\, ds + Z(s)\, dW(s), \quad Y(T) = h\bigl(t, X(T)\bigr), \tag{4.1}-\tag{4.2}$

Assuming a recursive cost structure, e.g., $$g(t, s, x, u, y, z) = -\delta y + g_0(t, s, x, u, z)$$, time-consistency in the tail is approached analogously to the Bolza case by matching the solution at $T_0$ with the infinite-horizon recursive value function.

5. Key Assumptions, Verification, and Implications

Key conditions:

• (H1): Regularity and uniform ellipticity/controllability of $b$, $\sigma_i$.
• (H2): Integrability and growth conditions on $g$ ensuring that $J^0(t, x; u(\cdot))$ is finite.

The splitting method relies on the assumption (1.15) (eventual exponential discount), which enables the "tail" problem to inherit all the properties of classical time-consistent stochastic control.

Time consistency is achieved in the tail because with exponential discount and possibly non-stationary, but regular, coefficients, the dynamic programming principle (DPP) holds: the BeLLMan equation yields optimal feedbacks that remain optimal irrespective of the starting time, as long as the augmented state fully characterizes the system.

6. Broader Context and Connections

The decomposition strategy parallels the general paradigm for handling time-inconsistency in stochastic control:

• The tail non-autonomous, time-consistent problem admits classical verification arguments and HJB analysis.
• The initial segment, inherently time-inconsistent, is treated via equilibrium or generalized (game-theoretic or fixed-point) methods.
• The resulting equilibrium strategy "locks in" the solution, precluding incentive to deviate when the tail problem is rejoined.

This approach extends to problems with recursive utilities (e.g., via BSDEs), and the theory is robust to inclusion of running and terminal costs, non-autonomous coefficients, and both degenerate and non-degenerate diffusions.

7. Conclusion

A non-autonomous time-consistent problem, as extracted from the tail of a decomposed infinite-horizon time-inconsistent control problem (Wei et al., 18 Sep 2025), is mathematically formulated as a stochastic optimal control with explicitly time-dependent coefficients and exponential discounting starting after a sufficiently large time. This framework ensures that the dynamic programming principle and time-consistent policies are applicable in the tail region, and serves as the anchor for constructing equilibrium solutions to the original, more general time-inconsistent stochastic control problem.