martingale techniques

This paper addresses quantitative model checking and controller synthesis for shift-invariant specifications (e.g., ω-regular properties, LTL). We propose a unifying framework based on martingale certificates. Methodologically, we establish the first theory for constructing martingales that either exactly compute (for finite-state systems) or arbitrarily tightly approximate (for general state spaces) the satisfaction probability, integrating stochastic invariants, convex optimization, and symbolic computation. Our key contributions are: (1) systematically extending classical “almost-sure” verification to yield quantifiable probabilistic bounds; (2) enabling unified treatment of diverse specifications—including reachability, safety, and stability—under a single certificate-based paradigm; and (3) achieving tight upper and lower bounds in multiple infinite-state case studies, thereby significantly enhancing both the expressiveness and practical applicability of martingale-based methods.

Existing martingale-based verification methods for discrete-time infinite-state stochastic systems support only quantitative reachability/safety or qualitative (probability-one) ω-regular properties, fundamentally limiting their applicability to general quantitative ω-regular specifications. Method: We propose the first verifiable martingale certificate framework for quantitative ω-regular properties, based on the product space of the system and a limit-deterministic Büchi automaton (LDBA). We introduce a novel *LDBA-based supermartingale (LDBSM)* certificate and develop a template-based automated synthesis algorithm grounded in polynomial inequality solving. Contribution/Results: Our approach enables fully automated verification and controller synthesis for general quantitative ω-regular properties—e.g., “visit a target set infinitely often with probability ≥ 0.95”—extending supermartingale techniques beyond prior capabilities. We demonstrate its effectiveness on polynomial dynamical systems, successfully solving multiple benchmark quantitative ω-regular problems previously intractable via supermartingale methods.

This paper investigates distributionally robust sensitivity analysis of model risk under martingale constraints—or equivalently, fixed first-order marginal distributions—in the Wasserstein space. We propose the first unified framework jointly modeling distributionally robust minimization and semi-static hedging, yielding explicit closed-form solutions for first-order optimal hedging strategies. Our methodology integrates Wasserstein probability metrics, martingale-constrained optimization, and semi-static derivative hedging theory, providing a unified characterization of robustness bounds under both standard and generalized Wasserstein distances. The main contributions are: (1) a novel paradigm for quantifying first-order sensitivity of model risk; (2) implementable, analytically tractable optimal semi-static hedging strategies; and (3) an extension of distributionally robust financial modeling to non-i.i.d., non-Markov, path-dependent settings—substantially enhancing robustness and practical applicability in real-world markets.

This paper addresses path-dependent distributionally robust stochastic control under non-concave loss functions over a finite horizon, aiming to mitigate model misspecification risk in extreme scenarios such as financial crises. We establish the first dynamic programming principle applicable to non-concave objectives and propose a unified, path-dependent ambiguity set framework compatible with both Wasserstein balls and parametric distribution families. Furthermore, we introduce a novel robust hedging paradigm for financial derivatives that explicitly accommodates bilateral, asymmetric preferences of buyers and sellers, and implement fully data-driven ambiguity set construction. Empirical results demonstrate that the proposed strategy significantly outperforms delta hedging and non-robust methods based on the empirical measure under extreme market conditions, markedly enhancing hedging robustness. Our core contributions lie in (i) a theoretical breakthrough—dynamic programming for non-concave robust optimization; (ii) methodological innovation—path-dependent and data-driven ambiguity modeling; and (iii) financial application advancement—robust hedging under asymmetric preference structures.

This paper identifies an error in a corollary of Theorem 2.8 in Bayraktar & Yu (2018), undermining their market viability conclusion under proportional transaction costs. To address feasibility, it proposes strict consistent local martingale systems (SCLMS) — replacing the conventional strict consistent pricing systems — as the dual criterion, and constructs a unified verification framework based on two weak no-arbitrage conditions: NUPBR (no unbounded profit with bounded risk) and the newly introduced NLABP (no local acceptable profit). It establishes, for the first time, the robust equivalence between SCLMS and both NUPBR and NLABP. The introduction of NLABP extends the scope of arbitrage-free theory to broader settings with transaction costs. Finally, the paper derives necessary and sufficient conditions for market viability under proportional transaction costs, providing a novel theoretical foundation for utility maximization in frictional markets.

This study addresses a linear-quadratic stochastic optimal control problem subject to state constraints, aiming to steer the system trajectory away from prescribed forbidden regions in space-time while minimizing the expected cost of state and control. By modeling the dynamics via diffusion processes and leveraging stochastic control theory together with probabilistic representation techniques, the authors establish a probabilistic representation of the value function under regularity conditions on the constraint set and derive its explicit solution. The resulting optimal control policy is strongly adapted and implementable via the filtration generated by the underlying Brownian motion. In addition to providing several analytical examples, this work offers a systematic framework for solving stochastic control problems with state constraints.

This work addresses the absence of rigorous formalization of Itô integration and Itô’s formula in existing proof assistants, particularly the lack of machine-verified treatment of these constructs as martingale processes. Building upon Lean 4, Mathlib, and the BrownianMotion library, we develop an L²-theoretic Itô calculus on a bounded interval [0,T] by constructing the Itô integral via Hilbert space isometry, establishing it as an L²-continuous martingale, and proving Itô’s formula for C³ functions with an explicit remainder bound. To our knowledge, this is the first machine-checked verification of Itô’s formula in any proof assistant and the first formalization of the Itô integral as a martingale-valued process. A single structural identity uniformly yields adaptivity, the martingale property, contraction bounds, and both forms of Itô isometry. The entire development comprises approximately 7,200 lines of sorry-free code across 22 modules, with all main theorems validated under classical axioms.

This work investigates the concentration of iteration errors in stochastic approximation algorithms driven by heavy-tailed Markov noise, covering both expansive and non-expansive operator settings. Under a framework involving a finite-state Markov component and martingale difference noise, the authors construct a novel Lyapunov function via the moment-generating function of the solution to the Poisson equation, complemented by auxiliary projection and black-box truncation techniques to reduce unbounded noise to a bounded setting. The study provides the first systematic characterization of the fine structure of error tails: under bounded noise, tails can be sub-Gaussian, sub-Weibull, or intermediate between Pareto and Weibull; under unbounded noise, if the operator is almost surely non-expansive, the error tail is at most three times heavier than that of the noise, whereas if the operator is expansive with positive probability, significantly heavier tails may arise, with sharp worst-case examples demonstrating the tightness of these bounds.

This work addresses the limited generalization of existing methods in complex scenarios by proposing a novel architecture based on adaptive feature fusion and dynamic inference. The approach effectively integrates local details and global semantic information through a multi-scale context-aware module and a learnable routing strategy, enabling the model to dynamically adjust its computational pathway during inference according to input content. Experimental results demonstrate that the proposed model significantly outperforms current state-of-the-art methods across multiple benchmark datasets while maintaining low computational overhead. The primary contribution lies in the introduction of a general and efficient dynamic inference framework, offering a new perspective for enhancing model robustness on out-of-distribution data.

This work addresses the finite-time convergence of stochastic iterative algorithms for fixed-point equations accessible only through a noisy oracle. The authors propose a norm-independent, unified Lyapunov function framework constructed via a generalized Moreau envelope, which integrates Lyapunov stability theory with stochastic approximation analysis. This framework accommodates complex settings such as Markovian noise, seminorm contractive operators, and dissipative operators, yielding sharp non-asymptotic convergence bounds in both high-probability and mean-square senses. As a result, it provides a unified and refined finite-time convergence guarantee for a broad class of algorithms, including stochastic gradient descent, linear stochastic approximation, Q-learning, and temporal difference learning.