value function approximation

This paper addresses error propagation in discrete-time stochastic optimal control under the dynamic programming (DP) framework, specifically analyzing how estimation errors accumulate and amplify during backward recursion. Method: We propose a natural decomposition of value function estimation errors, integrating kernel ridge regression in reproducing kernel Hilbert spaces with Monte Carlo simulation to rigorously characterize both per-step estimation errors and their backward propagation dynamics. Contribution/Results: We establish the first theoretical framework for error propagation analysis in DP, proving that accumulated error grows controllably with the number of backward steps. The method significantly improves accuracy and numerical stability in American option pricing—demonstrating robust convergence even in high-dimensional and non-Markovian settings. By unifying theoretical rigor with computational feasibility, our approach provides a novel paradigm for financial derivative pricing and policy optimization.

This study addresses the challenge posed by the “curse of dimensionality,” which renders conventional numerical methods ineffective for high-dimensional dynamic stochastic models in economics and finance. To overcome this limitation, the work proposes an innovative integration of deep learning and dynamic economic modeling by combining deep equilibrium networks, physics-informed neural networks, differentiable surrogate models, and Gaussian processes. Enhanced with active learning and dimensionality reduction techniques, the proposed framework efficiently solves heterogeneous-agent, macro-financial, and climate-economy models characterized by extremely large continuous state spaces. The approach not only facilitates structural estimation and policy simulation but also substantially improves computational efficiency and estimation accuracy. Broad applicability is further supported through open-source, reproducible code.

This work addresses the curse of dimensionality and poor cross-parameter generalization in approximating value functions for high-dimensional generalized/differential games with state constraints. We propose a Hybrid Neural Operator (HNO) that maps game parameters directly to the value-function space, integrating supervised data with physics-informed sampling from the full-space-time Hamilton–Jacobi–Isaacs (HJI) equation. To enhance robustness in safety-critical settings, HNO incorporates nonlinear dynamics embedding and a Lipschitz-aware training strategy. Compared to supervised neural operators (SNOs), HNO achieves superior safety performance under identical computational budgets in 9D and 13D nonlinear dynamical systems. Moreover, it enables real-time inference for human–machine and multi-agent interactions while maintaining convergence stability and constraint satisfaction.

This work proposes FPILOT, a novel framework that introduces model predictive control into financial reinforcement learning to overcome the limitations of static trading policies. Existing reinforcement learning agents typically employ fixed strategies during inference and cannot dynamically adapt based on price forecasts. In contrast, FPILOT leverages multi-step unconditional price predictions to construct return targets and performs real-time optimization of any pretrained policy at inference time—without requiring retraining. The approach is particularly effective in enhancing stochastic policies and achieves significant improvements in both cumulative returns and risk-adjusted performance metrics—including Sharpe, Sortino, and Calmar ratios—on the TradeMaster DJ30 benchmark. Moreover, the gains scale consistently with the quality of the underlying price predictions.

This paper addresses the efficient pricing of swing options in interruptible electricity markets. Within a backward dynamic programming framework, it focuses on high-accuracy approximation of the conditional expectation—i.e., the continuation value. Two function approximation approaches are proposed: least-squares regression and deep neural networks. The work establishes, for the first time, three rigorous theoretical results: (1) continuity of the swing value function with respect to cumulative consumption; (2) double convergence of both approximations as the model dimension (m o infty) and the number of Monte Carlo paths (N o infty), with an error bound of (O(1/sqrt{N})); and (3) uniform convergence along the path (V^{m,N} o V^m o V). These contributions yield a novel pricing paradigm for swing options that combines mathematical rigor with computational tractability.

This work addresses the long-standing challenge of efficiently representing and computing square-integrable predictable stochastic processes in $\mathcal{H}^2_T(\mathbb{R}^d)$, which has been hindered by the high-dimensional basis functions and intricate iterated integrals inherent in classical Wiener chaos expansions. We propose NeuralChaos, a novel neural operator architecture capable of generating processes that satisfy both predictability and square-integrability using only finitely many Brownian motion samples. Theoretically, we establish that NeuralChaos is dense in $\mathcal{H}^2_T(\mathbb{R}^d)$ and achieves the optimal $N$-term approximation rate with respect to chaoslet bases. Moreover, we show that compressible processes are generic, whereas finite-dimensional Markovian neural SDE models constitute a measure-zero, sparse subset. Integrating neural operators, Wiener chaos, Malliavin–Sobolev regularity, and non-degenerate sub-Gaussian sampling, our approach substantially enhances modeling expressivity and computational efficiency in stochastic dynamic control and dynamic hedging tasks.

This work addresses the limitation of existing Neural Jump ODE (NJ-ODE) methods, which are confined to finite-dimensional processes and thus unsuitable for directly modeling continuous-time stochastic processes in function spaces—such as yield curves. The paper presents the first extension of NJ-ODE to infinite-dimensional $L^2$ function spaces by integrating neural operators with the NJ-ODE framework, yielding an end-to-end operator-valued neural differential equation model. This approach learns the conditional expectation of functional-valued processes directly from discrete, irregularly sampled, and potentially incomplete observations, circumventing information loss due to spatial discretization. Under weaker assumptions than prior work, the authors establish $L^2$-convergence of the model to the true conditional expectation, thereby unifying and generalizing finite-dimensional theory while offering a novel paradigm for modeling high-dimensional dynamic objects like financial surfaces.

This work proposes a decision-theoretic neural pretraining framework to address key challenges in time series analysis, including finite-sample bias, poor calibration, and forecast combination. By jointly modeling the data-generating process and decision objectives within a simulated environment, the method trains neural networks via hierarchical simulation to approximate optimal decision rules, enabling high-quality zero-shot inference without real-world data. The approach innovatively integrates decision theory with deep learning, allowing explicit control over risk, bias, minimax performance, and calibration consistency, thereby effectively solving problems that are analytically intractable or computationally prohibitive. Empirical results demonstrate substantial improvements over conventional methods such as maximum likelihood estimation and AICc in AR(p) modeling and forecast combination tasks, while achieving competitive or superior performance against state-of-the-art statistical and deep learning models on real-world benchmarks.

This work addresses the challenges in high-dimensional discrete-time dynamic programming where recursive utility lacks a closed-form expression and the certainty equivalent in the Bellman equation is computationally intractable. The paper proposes a Certainty Equivalent Learning (CEL) algorithm that, for the first time, integrates deep learning into the recursive utility framework. By jointly approximating the value function, policy function, and certainty equivalent function with neural networks, CEL operates without grids, Euler equations, or differentiability assumptions on the state transition dynamics. As a purely simulation-based, mesh-free solver, the method achieves high-accuracy approximations across several high-dimensional economic and financial models, yielding out-of-sample Bellman errors and first-order condition residuals on the order of 1e⁻⁴ to 1e⁻³.

This work proposes a unified mathematical framework grounded in dynamic information flow for constructing structurally rigorous models of future prediction. By integrating filtering theory, regular conditional probabilities, Markov semigroups, infinitesimal generators, and multiple information geometries—including Hilbert, Fisher–Rao, and Wasserstein—the approach conceptualizes prediction as the construction of conditional distributions governed by informational, geometric, and modeling constraints. The framework elucidates deep connections among classical results such as the tower property and semigroup laws, as well as Itô’s formula and backward equations. Explicit transition laws, spectral decompositions, term structures, and asymptotic behaviors are derived within canonical models like Ornstein–Uhlenbeck and Cox–Ingersoll–Ross, thereby establishing a compact mathematical mapping from idealized theoretical constructs to empirical forecasting.