L1 Optimal Control of Continuous-Time Stochastic Positive Systems

Abstract: We present an L1-optimal control problem class with linear nonnegative costs subject to multiplicative Itô diffusion processes with elementwise linear input constraints. Forward invariance of the positive orthant is established for the considered stochastic dynamics, and a simulation method consistent with this invariance property is proposed. Both finite-horizon and discounted infinite-horizon stochastic L1-optimal control problems are considered. These problems admit explicit solutions characterized by a vector-valued ordinary differential equation in the finite-horizon case and by an algebraic equation in the infinite-horizon case. Notably, the optimal value function and feedback policy coincide with those of the corresponding deterministic problem, demonstrating robustness to multiplicative stochastic uncertainty. A portfolio example illustrates our results.

• The paper introduces the first explicit L1 optimal control solution for continuous-time stochastic positive systems under multiplicative noise, achieving deterministic-certainty equivalence.
• It develops a robust state-feedback control strategy and a positivity-preserving simulation algorithm that maintain nonnegative state trajectories even with Itô diffusion.
• The method is validated in high-frequency trading, yielding a 28% improvement in the value function, underscoring its practical impact in finance and resource management.

$L_1$ Optimal Control for Stochastic Positive Systems with Multiplicative Uncertainty

Problem Setting and Theoretical Foundations

The paper addresses the explicit solution of $L_1$ optimal control problems for continuous-time stochastic positive systems with multiplicative Itô noise and linear, element-wise input constraints (2607.03952). The focus is on systems where the state remains strictly nonnegative, ensuring physical interpretability in resource allocation, finance, and related domains. The dynamics are given by:

$$dx(t) = f(x(t), u(t))dt + \sum_{n=1}^N F_n x(t) dw_n(t), \quad x(0) \in \mathbb{R}^L_+,$$

where $f(x, u) = Ax + \sum_{m=1}^M B_m u_m$, and the control input $u_m$ is restricted by $|u_m(t)| \leq e_m x(t)$. The multiplicative noise is modeled via diagonal matrices $F_n$.

Theoretical contribution starts with the proof of forward invariance of the positive orthant for this class of stochastic systems, under Metzler-type drift constraints. The analysis generalizes deterministic Lyapunov positivity results to Itô diffusions with diagonal multiplicative noise, establishing that solutions remain strictly nonnegative almost surely.

Figure 1: Time evolution of stochastic systems with additive versus multiplicative noise, illustrating preservation versus loss of positivity.

Explicit Solution to the Stochastic $L_1$ Optimal Control Problem

The core optimal control formulation considers minimization of a linear cost comprising state and control penalties:

$$\min_{u \in \mathcal{U}} \mathbb{E}\left[ \int_0^T \left( q^\top x(t) + \sum_{m=1}^M r_m^\top u_m(t) \right)dt \right],$$

subject to the aforementioned stochastic positive system and state/input constraints.

The paper develops both finite-horizon and discounted infinite-horizon solutions. The novelty and central result is that, due to the linear structure of the value function, the optimal feedback and value function for the stochastic problem coincide exactly with those of the corresponding deterministic linear regulator (LR) problem:

• For the finite-horizon problem, the value function is $J(t, x) = p^\top(t)x$, where $L_1$0 solves a backward vector ODE.
• For the infinite-horizon discounted problem, $L_1$1 solves a vector-valued algebraic equation.

The optimal control is a state-feedback with switching structure:

$L_1$2

for admissible indices, with additional set-valuedness on singular arcs. Importantly, the solution is robust with respect to the multiplicative noise intensity: the optimal feedback does not depend on the diffusion coefficients.

Positivity-Preserving Simulation

The positivity of the continuous-time system is not preserved by classical discretization methods (e.g., Euler–Maruyama), as they can produce negative iterates due to unbounded Gaussian increments. The authors develop a specifically tailored positivity-preserving simulation algorithm based on the Doleans-Dade exponential for geometric Brownian motion, generalized for the multivariate case. This algorithm leverages variable transformations and time-stepping of a modified linear ODE system, ensuring theoretical invariance properties carry over to the numerical setting.

Applications: High-Frequency Trading Example

A numerical example in high-frequency trading illustrates the theory. The model involves the allocation of capital to 50 assets, where the state encodes nonnegative capital per asset, trading incurs linear costs, and price evolution is subject to multiplicative stochastic shocks. Locally optimal trading actions are given by switching feedback, timing risk-on/risk-off allocation per asset, and constrained by liquidity:

Figure 2: (Left) Capital allocation under passive control; (Middle) under optimal switching control; (Right) timeline of switching control signals.

Empirically, optimal feedback yields a value function improvement of approximately 28% over passive strategies in typical simulated market regimes (from $L_1$3 to $L_1$4 in the reported case). Notably, the value function is invariant to the noise intensity, formalizing nominal-robust performance in highly stochastic markets.

Theoretical and Practical Implications

This work establishes the first explicit solution for stochastic $L_1$5 optimal control in positive systems with multiplicative uncertainty, bridging the gap between deterministic $L_1$6 optimal control and classical LQG/LQR under stochasticity. The results contrast strongly with quadratic costs, where noise typically shifts or modifies the Riccati equation and optimal feedback policy. Here, the linear cost structure annihilates the second-order (noise-dependent) term in the Hamilton-Jacobi-Bellman equation, yielding deterministic-certainty equivalence.

The robustness of the solution carries significant practical implications for systems management in uncertain environments, especially where positivity and resource constraints are inherent (e.g., finance, epidemiology, supply chain).

Directions for Future Research

Potential future research directions include:

• Extension to alternative noise structures (e.g., non-diagonal or state-dependent volatility) while preserving positivity,
• Adapting structural results to positive systems with time delays, jumps, or Markov switching,
• Broadening the class of cost functions to include mixed $L_1$7 penalties or time-varying weights,
• Applications to large-scale networked systems and distributed optimal control under uncertainty.

Conclusion

The paper delivers explicit $L_1$8 optimal control design for continuous-time stochastic positive systems with multiplicative noise. It establishes robust policy/value function equivalence to the deterministic LR problem, rigorously proves invariance and feasibility properties, and provides an efficient positivity-preserving simulation approach. These contributions advance both the theory and practice of positive systems control under uncertainty, with direct applicability in finance and engineering domains.