Error Propagation in Dynamic Programming: From Stochastic Control to American Option Pricing

arXiv:2509.20239 · stat.ML, cs.LG, q-fin.CP, q-fin.PR, stat.AP · Submitted 2025-09-24 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: I'm Tom, and with me are Jane, Lu, senior AI researcher at Tsinghua, Meng, lead engineer at a mysterious AI startup and Lalam, the in-house Large Language Model.

Jane: Today's paper: "Error Propagation in Dynamic Programming".

Tom: This paper investigates theoretical and methodological foundations for stochastic optimal control (SOC) in discrete time, developing a framework to rigorously analyze how errors propagate backward through dynamic programming approximations.

Jane: First, who's behind it and why it matters.

Title and authors: Tom: Alright team, we’re moving into the section where they introduce the paper "Error Propagation in Dynamic Programming: From Stochastic Control to American Option Pricing," and we’ll talk about who wrote it and what they are trying to achieve.

Jane: It starts by setting up the mathematical context, explaining that this paper is looking at stochastic optimal control in discrete time, which essentially means making decisions over a sequence of steps where things change randomly.

Lu: The authors are Andrea Della Vecchia and Damir Filipovic from EPFL - Swiss Finance Institute (SFI) Lausanne, who bring expertise in both the mathematical finance side and the theoretical machine learning side.

Meng: I’m interested in what they mean by this combination of backgrounds; is it a purely theoretical exercise, or are they aiming for something directly applicable to real-world decision-making systems?

Lalam: From an AI perspective, having authors with that specific blend of expertise suggests they're aiming to bridge the gap between high-level mathematical control theory and the practical need for robust estimation in complex environments.

Tom: That’s exactly what this paper is doing, Jane; it’s not just abstract math. They are taking a general control problem and showing how to apply tools from modern machine learning, like kernel methods, to get answers that are useful in finance.

Jane: So they are taking the Bellman equation structure—which defines optimal decisions over time—and trying to solve it when the state space is too big for traditional exact solvers.

Lu: They specifically address the difficulty that well-established numerical methods like tree-based approaches or PDE solvers run into when complexity increases because of things like the curse of dimensionality in high dimensions.

Meng: That curse is a real headache for any engineer trying to build a robust planning system, so overcoming that limitation through approximation techniques is definitely worth focusing on.

Lalam: The authors are tackling this by proposing a specific sequence of approximations—combining nonparametric regression and Monte Carlo subsampling—to manage the complexity inherent in those high-dimensional settings.

Tom: And that’s the core of their approach, Lu; they aren't just throwing a standard deep learning model at it; they have a specific mechanism designed to handle the sequential nature of dynamic programming.

Jane: It sounds like they are building a specialized estimator tailored specifically for problems where decisions depend on the entire history, but in a compressed form.

Lu: They mention that while many problems are naturally Markovian, dimensionality constraints often force us to compress the history Z zero:t, u zero:t into just X t, which is where their Markov assumption comes into play.

Meng: So they’re making a trade-off between having a perfectly complete picture of everything and being able to actually compute something in a reasonable amount of time.

Lalam: That trade-off is very relevant for building scalable AI systems that need to make decisions quickly, so managing that compression is key for practical deployment.

The paper's summary: Tom: Now we’re looking at the actual substance of this paper, and they summarize their main contribution by detailing how they tackle the problem of error accumulation in approximate dynamic programming.

Jane: Essentially, the summary explains that they formulate a general dynamic programming framework to define the control objective over a time horizon T, and then show how their value function is estimated using approximations involving nonparametric regression within RKHSs and Monte Carlo sampling.

Lu: They lay out this decomposition of total error into three distinct parts: regression error, Monte Carlo sampling error, and propagation error that they rigorously control at each time step.

Meng: That systematic breakdown of the total error is what I find most interesting from a practical viewpoint; it tells us exactly which part of our approximation—the function fitting or the sampling—is causing the biggest issue.

Lalam: Lalam sees this decomposition as a huge win for AI culture because it shifts our focus from just getting *an* answer to understanding *why* that answer has an error, allowing for targeted improvements.

Tom: They go on to prove how these errors propagate backward in time-from maturity to the initial stage, which is that relatively underexplored aspect of the paper.

Jane: So they are showing us a principled way to analyze how noise introduced at an early step impacts the final decision made at time zero, which is a really deep dive into the mathematics of sequential learning.

Lu: They show that this propagation can be bounded using terms like Term I: Regression Error related to sample size n t and beta t+one and Term II: Monte Carlo Error controlled by empirical Rademacher complexity, which is then managed with concentration inequalities <ref:2509.20239#pg0>.

Meng: The explicit bounds on the regression error showing dependence on n t are what I need to hear; that tells me how much data we actually need to feed the system to get a good result.

Lalam: And they show that for finite control sets, the Monte Carlo error can be bounded by terms involving K M t, which is a clean way to quantify how fast our sampling noise shrinks as we use more samples.

Tom: And finally, they nail the propagation error term, bounding it as T T W lambda t+one t+one - T V t+one squared L squared mu t = c P E t+one which shows the error from step t is directly proportional to the error from step t+one <ref:2509.20239#pg0>.

Jane: It’s a beautiful structure, really; they show that by combining these specific learning rates, we can derive explicit convergence rates for the error at time t, which is what makes this paper so strong mathematically.

Lu: They conclude by selecting learning rates like lambda t about n-one beta t+one and M t about n beta t beta t+one which leads to the final result that the error at time t is bounded by Et A one/n T T + c P E t+one for t from zero to T-one.

Meng: That explicit formula for the error bound at every single time step is what translates this theory into a usable tool for iterative AI training loops.

Lalam: This paper really helps improve our culture by showing that in complex sequential AI, we can move away from just hoping the process converges and instead have a mathematical roadmap to manage the inevitable accumulation of errors.

The paper's improvements: Tom: Moving on to what they suggest for improvement, the authors focus on how we can actually use these results better in practice when applying this framework, especially concerning those convergence guarantees.

Jane: They suggest selecting specific learning rates—like lambda t about n-one beta t+one and M t about n beta t beta t+one —to ensure the recursion is contractive when the risk-free interest rate is strictly positive, which helps dampen errors.

Lu: That condition where c P < one makes the recursion contractive, which means that once you get a good estimate at a certain point in time, subsequent steps will naturally refine that estimate rather than drifting away <ref:2509.20239#pg0>.

Meng: From an engineering perspective, knowing that we need to tune those parameters based on the smoothness of the problem—the beta t terms—gives us a clear roadmap for allocating our computational resources efficiently.

Lalam: Lalam thinks this is fantastic because it moves us away from just picking arbitrary settings; it provides a data-driven way to choose the right balance between computation and accuracy.

Tom: And they also suggest that when applying this framework, we should look at how the total error is decomposed into regression error, Monte Carlo error, and propagation error to diagnose whether our performance issues are due to model fitting or sampling inaccuracies.

Jane: That diagnostic capability is huge; it means if the regression term is large, we need more data or a better kernel function, but if the propagation term dominates, we need to focus on stabilizing earlier steps.

Lu: They also suggest that for high-dimensional scaling, they look toward techniques like random projection methods within the Kernel Ridge Regression step—like the Nystrom method or FALKON algorithm—to keep things computationally feasible for very large inputs.

Meng: Scaling up to handle dozens of assets or very detailed sensor data is definitely the next frontier, so integrating these projection techniques into the regression step makes this framework much more viable for real-world systems.

Lalam: This points toward a future where AI systems can be both theoretically sound and practically scalable, because they’re designing the architecture to handle massive complexity without losing control over the error propagation.

Conclusion: Tom: So we’ve covered a lot on "Error Propagation in Dynamic Programming: From Stochastic Control to American Option Pricing," summarizing how this paper provides a detailed framework for bounding approximation errors by decomposing them into regression, sampling, and propagation components across time steps.

Jane: The main idea is that the authors show us a principled way to analyze how approximations made at one step affect future decisions, leading to explicit convergence rates based on carefully chosen learning rates.

Lu: The paper confirms that this approach works for pricing American options and provides a solid mathematical justification for using hybrid Monte Carlo and kernel methods in this sequential control setting.

Meng: It gives us concrete benchmarks on how much data is needed to achieve a certain level of accuracy, which is extremely useful for our engineering roadmap when we have to decide how much simulation time to invest.

Lalam: Lalam feels that the ultimate impact here is showing that we can build sequential AI systems where we have a mathematical roadmap for managing the inevitable accumulation of errors, making the whole process more reliable.

Tom: And as they conclude, they emphasize that when using this paper’s framework correctly, especially when r > zero the recursion becomes contractive, which means errors naturally shrink over time.

Jane: It’s a really strong conclusion because it moves from just describing the math to providing actionable advice on how to tune the system for practical performance in high-dimensional control problems.

Lu: The implication is that we can tackle those complex planning tasks with more confidence because we have a structured way to manage uncertainty, which is what makes this work so interesting.

Meng: I think the real takeaway for us is that we have a proven methodology now for balancing accuracy and computational cost in sequential decision making.

Lalam: Lalam concludes that this paper contributes a vital tool to the AI community by providing a rigorous way to control error propagation in dynamic programming, which will help build more trustworthy and effective sequential AI systems across the board.

Andrea Della Vecchia, Damir Filipovic

EPFL - Swiss Finance Institute (SFI)

stat.ML, cs.LG, q-fin.CP, q-fin.PR, stat.AP

Submitted: 2025-09-24

Updated: 2026-10-02

Code: https://github.com/FalkonML/falkon

Importance score: 61/100

The gist: This paper investigates theoretical and methodological foundations for stochastic optimal control (SOC) in discrete time, developing a framework to rigorously analyze how errors propagate backward

Key concepts

Stochastic Optimal Control (SOC)
This is a problem where you need to make sequential decisions over time to maximize a reward while dealing with random uncertainties. The goal is to find the best control strategy, which in this paper involves using dynamic programming approximations to estimate the optimal value function.
Dynamic Programming (DP) Recursion
DP is a method that solves complex problems by breaking them down into smaller, overlapping subproblems. In this context, it means calculating the optimal value function at each time step based on the solution from the next time step. The core equation relates the current value to future values.
Error Decomposition
The total error in approximating an optimal solution is broken down into three parts: regression error (how well a function fits data), Monte Carlo sampling error (error from using random samples instead of true expectations), and propagation error (error passed from the previous time step). This helps pinpoint where the approximation is failing.
American Option Pricing
This applies the control framework to financial problems, specifically pricing American options. An American option allows exercise at any time before maturity. The paper uses this structure to show that its proposed method can accurately price these complex derivatives.

Terminology

Summary

This paper investigates theoretical and methodological foundations for stochastic optimal control (SOC) in discrete time, developing a framework to rigorously analyze how errors propagate backward through dynamic programming approximations. The gist: A general RKHS-based formulation of approximate dynamic programming is proposed, providing a rigorous decomposition of total approximation error into regression error, Monte-Carlo sampling error, and propagation error.

Problem Formulation and Dynamic Programming

The work starts by formulating the control problem in a general dynamic programming framework for discrete time horizon t = 0 to T. The objective is to maximize the sum of partial rewards Ft over time plus a terminal reward Φ at time T, leading to the optimal value function Vt defined by the Bellman equation:

**/Vt (x) = sup u∈U E "T X−1 s=t Fs (Xu s, us(Xu s)) + Φ (Xu T) Xu t **

The dynamic programming recursion is given by:

**/Vt(x) = TtVt+1(x), t ∈ 0,..., T − 1. **

To solve this recursion in high-dimensional settings, the paper introduces a sequence of approximations combining nonparametric regression methods and Monte Carlo subsampling to estimate the associate value function.

Sample-Based Value Function Approximation

The stochastic dynamic control problem is approximated using empirical Bellman operators derived from Monte Carlo simulation. The process involves:

  1. Generating samples: Let Mt i=1 ∼ P Mt t+1 be i.i.d. samples from the distribution of the stochastic driver Zt+1.

  2. Defining the empirical operator: P ‹u t f(x):= 1/Mt X Mt i=1 f Ä πt(x, u, z (t+1) i) ä, T ‹tf(x):= ess sup u∈Ut ¶ Ft(x, u) + P ‹u t f(x)

  3. Using regression: A sequence of function approximators Wλt+1 is constructed by solving a supervised learning problem where the target function is the empirical continuation value: yi = T‹tWλt+1 t+1 (xi).

Error Decomposition and Propagation Analysis

A rigorous analysis focuses on bounding the total error Et = Wıλt t − Vt 2 L2 µt. This total error is decomposed into three distinct components:

/Term I: Regression Error.

This term corresponds to the excess risk of Wıλt t, which is bounded using the source condition (Assumption 3) and kernel methods, yielding a rate of convergence dependent on sample size nt: Wıλt t − T‹tWλt+1 t+1 2 L2 µt ≲ n − βt βt+1 t.

/Term II: Monte Carlo Error.

This accounts for the error in approximating the unknown expectation, bounded by terms involving the empirical Rademacher complexity ER“(F x t),” which is controlled using concentration inequalities. For finite control sets, this can be bounded as: ER“(F x t) ≲ log K Mt.

/Term III: Propagation Error.

This term captures the error inherited from the previous step, bounded by: TtWıλt+1 t+1 − TtVt+1 2 L2 µt = cP Et+1.

Final Convergence Guarantees

By combining these bounds and selecting appropriate learning rates (e.g., λt ∼ n− 1 βt+1 and Mt ∼ n βt βt+1 t), the paper derives explicit convergence rates. The final result for the error at time t is: Et ≲ Å 1/ntã βt βt+1 + cP Et+1, for t ∈ 0,..., T − 1. For the initial value function estimate V0, this leads to: E0 = Wıλ0 0 − V0 2 L2 µ0 ≲ T X−1 t=0 c t P Å 1/ntã βt βt+1. The analysis shows that when the risk-free interest rate r is strictly positive (cP < 1), the recursion becomes contractive, damping errors.

Application to American Options

The framework is applied to American option pricing, which is formulated as a finite-horizon optimal stopping problem. The transition function πt(x, u, z) encodes the dynamics of the underlying asset process under control decisions (exercise or hold). The algorithm proposed in Algorithm 1 demonstrates that the KRR-DP method performs competitively with benchmarks like GPR-Tree and GPR-MC for pricing geometric basket put and max-call options. The numerical simulations confirm that the method offers a favorable trade-off between accuracy and computational efficiency.

Improvements for AI systems

Based on the provided scientific paper, here are specific, high-impact improvements for AI systems that could be derived from its methodology:

) Improvements for AI Systems Derived from this Paper

The core contribution of this paper is a rigorous framework for solving Stochastic Optimal Control (SOC) problems in discrete time using a hybrid approach combining Monte Carlo simulation and Kernel Ridge Regression (KRR). This framework provides provable error bounds and handles the propagation of approximation errors backward through time.

Here are specific improvements for AI systems:

  2. High-Dimensional Sequential Decision Making Under Uncertainty: The system can now tackle complex, multi-stage decision problems where the state space is high-dimensional (e.g., portfolio management with many assets, robotics path planning in cluttered environments). Unlike standard Deep Reinforcement Learning (DRL) which often struggles with long horizons or complex dynamics due to the curse of dimensionality, this system uses a backward induction framework combined with RKHS regression to maintain tractability.

  4. Provable Accuracy and Trustworthy Predictions: The system moves beyond purely empirical methods by providing explicit error bounds (Theorem 1). This means that when the AI predicts an optimal action or value function, the user can quantify exactly how much confidence they should have in that prediction based on the sample size and model misspecification. This is crucial for high-stakes applications like financial risk management or autonomous vehicle control where black-box accuracy is insufficient.

  6. Adaptive Sample Efficiency: The system employs a sophisticated sampling strategy (Algorithm 1) that intelligently allocates computational resources between Monte Carlo simulation (for continuation value estimation) and Kernel Ridge Regression (for function approximation). The sample sizes for both components are dynamically tuned based on the time step's smoothness parameters, leading to superior performance compared to naive recursive methods. This makes the system significantly more data-efficient, requiring fewer samples to achieve a target accuracy.

  8. Financial Derivative Pricing with Rigorous Guarantees: The system can price complex American or Bermudan options (as demonstrated in Example 1 and Example 2) with theoretical convergence guarantees, rather than relying solely on heuristic Monte Carlo methods (like standard Longstaff-Schwartz). This allows for more robust valuation of exotic derivatives where the optimal stopping boundary is complex.

  10. Model Misspecification Robustness: The framework explicitly decomposes total error into regression error, Monte Carlo error, and propagation error (Section 4). By analyzing the Source Condition (Assumption 3), the system can quantify how sensitive its final solution is to errors in the underlying model or approximation of the value function. This allows researchers to diagnose whether performance degradation is due to poor data/sampling or poor functional form selection.

  12. Goal-Based and Policy-Based AI: The discrete-time SOC formulation (Eqs. 5, 6, 7) naturally extends to learning policies for goal-based tasks (e.g., robotic control where the reward is defined by reaching a specific state). This allows AI to learn sequences of actions that maximize a cumulative reward over time, even in non-Markovian settings (if the state representation is augmented).

  14. Scalable High-Dimensional Control: By leveraging techniques like random projection (Nystrom method, FALKON algorithm) within the KRR step, the system can be scaled to handle very high dimensions (e.g., 20+ assets or high-resolution sensor data) while maintaining computational feasibility, bridging the gap between theoretical rigor and practical scalability.

Sources

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