DeepMartingale: Dual Neural Approach

• DeepMartingale is a deep learning framework that integrates martingale duality with neural network parameterization to tackle high-dimensional optimal stopping problems.
• It approximates the integrand of the dual Doob martingale to compute tight upper bounds with proven convergence, avoiding the curse of dimensionality.
• The approach is practically applied for pricing options and optimizing risk management by simulating and controlling stochastic processes in financial markets.

DeepMartingale refers to a class of deep learning frameworks and architectures that leverage the martingale representation and duality principles to address high-dimensional optimal stopping problems by learning martingale structures—especially in the context of financial mathematics, risk management, and stochastic control. The DeepMartingale methodology constructs a neural approximation of the dual Doob martingale (or equivalently its integrand) associated with the value process, establishing tight upper bounds for the value function with provable convergence guarantees and rigorously analyzed expressivity that avoids the curse of dimensionality (Ye et al., 13 Oct 2025).

1. Foundational Principles

Optimal stopping problems seek to maximize the expected payoff $\mathbb{E}[g(\tau, X_\tau)]$ by finding a stopping time $\tau$ adapted to a filtration generated by the underlying process $(X_t)_{t\geq0}$. Classical (primal) solutions iteratively compute or approximate the Snell envelope (the value function) by dynamic programming or backward induction. In contrast, the dual approach, which underpins DeepMartingale, reframes the value as an infimum over a family of martingales, grounded in the Doob–Meyer decomposition: $Y^*_n = Y^*_0 + M^*_{t_n} - A^*_{t_n}$ where $(M^*_t)$ is a martingale and $(A^*_t)$ is an increasing, predictable process. The unique Doob martingale admits representation through the stochastic integral

$M^*_{t_n} = \int_0^{t_n} Z^*_s \, dW_s$

with $Z^*_s$ being the integrand process, typically functional of the observed filtration, and $W$ denotes the driving Brownian motion.

The dual formulation expresses the value function as

$$Y^*_n = \inf_{M \in \mathcal{M}} \mathbb{E} \left[ \max_{n \leq m \leq N} \big( g(t_m, X_{t_m}) - M_{t_m} + M_{t_n} \big) \,\big|\, \mathcal{F}_{t_n} \right].$$

By focusing on neural approximation of $Z^*$ (and thus $M^*$), DeepMartingale enables efficient sampling-based algorithms for dual bounds.

2. Martingale Representation and Neural Parameterization

DeepMartingale implements the martingale representation theorem at the algorithmic level. The process $M^*$ is discretized over a time mesh with subinterval points $t^n_k$, and the optimal integrand $Z^*_n(t^n_k, x)$ is expressed as

$$Z^*_n(t^n_k, x) = \frac{1}{\Delta t^n_k} \mathbb{E}\left[ Y^*_{n+1} \Delta W_{t^n_k} \mid X_{t^n_k} = x \right].$$

Deep neural networks $z^{\theta_n}_n(t,x)$ serve as universal approximators for $Z^*_n$, mapping the state-time pair to the candidate integrand. The discrete-time martingale approximation is then

$$\hat{M}_{t_{n+1}} = \hat{M}_{t_n} + \sum_{k} z^{\theta_n}_n(t^n_k, X_{t^n_k}) \cdot \Delta W_{t^n_k}.$$

This procedure leverages the conditional structure of the value process and enables scalable simulation-based optimization.

3. Duality, Upper Bounds, and Convergence

Within the DeepMartingale paradigm, the neural network–parameterized $\hat{M}$ is used to construct an explicit dual estimator: $$\tilde{U}_n(\hat{M}) = \max_{n \leq m \leq N} \left( g(t_m, X_{t_m}) - \hat{M}_{t_m} + \hat{M}_{t_n} \right).$$ The upper bound property

$Y^*_n \leq \mathbb{E}[\tilde{U}_n(\hat{M})]$

holds for any candidate $\hat{M}$. Theoretical analysis in DeepMartingale establishes that, under mild regularity conditions (e.g., Lipschitz continuity and growth controls for payoff $g$ and diffusion $X$), as the partition is refined (number of subintervals increases) and the neural approximation improves, the estimator converges in mean square to the true value: $$\mathbb{E} \left| \tilde{U}_n(\hat{M}) - Y^*_n \right|^2 \to 0.$$ This result ensures that DeepMartingale produces both tight and reliable upper bounds for the primal value.

4. Expressivity and Complexity Bounds

A central theoretical advance is the rigorous analysis of neural network expressivity. For any prescribed error $\varepsilon>0$, there exists a network architecture parameterizing $Z$ such that the resulting estimator approximates the true value function within $\varepsilon$ accuracy. Significantly, network size satisfies

$$\text{size} \leq \tilde{c}\, D^{\tilde{q}} \varepsilon^{-\tilde{r}},$$

where $\tilde{c}, \tilde{q}, \tilde{r}$ are universal constants independent of the state dimension $D$ and the inverse accuracy $\varepsilon^{-1}$. Thus, DeepMartingale avoids the curse of dimensionality—computational complexity grows at most polynomially with dimension and inverse accuracy, not exponentially. This expressivity derives from the ability of neural networks to approximate high-dimensional conditional expectation maps (via the universal approximation theorem) under affine Itô or related diffusive dynamics.

5. Numerical Methods and Stability

DeepMartingale's numerical implementation divides each time interval $[t_n, t_{n+1}]$ into $N_0$ subintervals, enabling a piecewise-constant approximation of $Z^*$ and simulating stochastic integrals via discretized Brownian increments. Training is performed by minimizing empirical surrogates to

$\mathbb{E}[\tilde{U}_n(\hat{M})],$

using stochastic gradient descent and Monte Carlo sampling.

Empirical results include tests on Bermudan options (max-call and basket-put) with both symmetric and asymmetric setups. The DeepMartingale estimator was consistently closer to the true value, displayed smaller statistical variances, and was robust to increases in state space dimension. This confirms the theoretical properties of convergence and expressivity, and supports claims regarding stability and avoidance of high-variance estimations.

6. Practical Implications and Applications

DeepMartingale's approach is particularly well-suited for computational finance and risk management:

• Pricing of Bermudan/American options: The method can efficiently compute tight upper bounds for options with early exercise features in high-dimensional models, supporting complex hedging and risk analysis scenarios.
• Production and operational decision-making: Stochastic models for machine activation, inventory, or investment decisions often reduce to high-dimensional optimal stopping problems, benefiting from DeepMartingale's scalability.
• Hedging strategy construction: The neural representation of the integrand $Z$ directly furnishes approximate optimal hedging strategies, aiding risk mitigation under incomplete market models.

A plausible implication is that DeepMartingale could extend to other stochastic control settings requiring martingale-based duality, such as robust optimization and reinforcement learning under uncertainty.

7. Theoretical and Methodological Impact

DeepMartingale unifies martingale representation, stochastic control theory, and deep neural approximation within a rigorous duality-based framework. It demonstrates that with careful architectural design, neural network–based methods can avoid both the exponential complexity of traditional numerical approaches and the instability/variance issues of naive Monte Carlo dual methods. The concrete complexity bound, independence of dimension, and empirical validation position DeepMartingale as a benchmark for future research in stochastic control and mathematical finance.

The method illustrates how deep learning, when guided by structural stochastic properties (such as martingale representation), not only offers practical computational solutions but also advances theoretical understanding regarding approximation power in high-dimensional probabilistic systems (Ye et al., 13 Oct 2025).