girsanov theorem application

This work addresses the long-standing open problem of the lack of quantitative characterization for the Robbins–Siegmund theorem by establishing, for the first time, a precise quantitative version of its Tao-style metastability. Methodologically, it innovatively introduces the metastability analysis framework into stochastic optimization theory, developing a metastable Doob decomposition and an $L_1$-supermartingale propagation technique, combined with refined probabilistic inequalities and quantitative estimates for stochastic processes. The main contributions are: (i) derivation of a universal upper bound on the metastable convergence radius, rigorously quantifying the number of iterations required for an algorithm to enter a stable region; and (ii) provision of the first computable and verifiable convergence guarantee for algorithms—such as SGD and stochastic approximation—that rely fundamentally on the Robbins–Siegmund theorem, thereby bridging a critical gap between classical asymptotic convergence theory and practical algorithmic verification.

This paper addresses the problem of constructing a measure-preserving martingale interpolation between prescribed marginal distributions in arbitrary dimensions, such that the resulting process is as close as possible to Brownian motion. Departing from prior work—largely restricted to one dimension and requiring finite second moments—we propose the first algorithm applicable under the significantly weaker assumption that marginals possess finite moments of order strictly greater than one. Our method establishes a novel connection between Bass martingales and semimartingale optimal transport, leading to the Measure-Preserving Martingale Sinkhorn (MPMS) algorithm. MPMS is derived from a dual formulation and integrates stochastic analysis, optimal transport, and fixed-point theory. We rigorously prove its monotonic convergence and dimensional scalability. Empirical evaluations on both synthetic and real financial datasets demonstrate that MPMS consistently outperforms existing approaches in accuracy and stability.

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 the conditional distribution of Banach space-valued jointly Gaussian random variables. It establishes that the conditional distribution remains Gaussian and develops a finite-dimensional approximation scheme based on Banach space-valued martingales to compute the conditional mean and covariance operator exactly. Methodologically, it unifies nuclear norm convergence and weak convergence analyses—yielding, for the first time, a rigorously convergent Gaussian conditioning theory in general Banach spaces. The framework applies broadly, including to reproducing kernel Hilbert spaces (RKHS) and spaces of continuous functions. For continuous Gaussian process paths, it guarantees uniform convergence of the conditional mean and covariance functions, as well as weak convergence of the conditional probability measures. These results provide a rigorous, general mathematical foundation for infinite-dimensional statistical inference and Bayesian inverse problems involving Gaussian processes.

Discrete diffusion models suffer from a lack of systematic theoretical error analysis, limiting their accuracy and reliability in generative modeling. To address this, we introduce the first unified analytical framework for discrete diffusions based on Lévy-type stochastic integrals, establishing— for the first time—the stochastic integral representation of discrete diffusion processes and proposing a Girsanov-type measure transformation theorem. Building upon this foundation, we derive the first explicit KL-divergence error bound for the τ-leaping discretization scheme. Our framework bridges the theoretical gap between discrete and continuous diffusion models, explicitly characterizing error sources—including state dependence of intensity functions and approximation bias from jump truncation. It unifies and strengthens existing convergence and stability results, providing a rigorous mathematical foundation and practical design principles for developing efficient, verifiable discrete diffusion algorithms.

This work addresses the inconsistency between training and inference in existing speculative decoding methods, where training optimizes only a single greedy path while inference requires verifying multiple sampled paths. To bridge this gap, we introduce variational inference into speculative decoding for the first time, reformulating draft model training as posterior inference over latent proposal paths by maximizing the marginal probability of acceptance under the target model. We propose a path-level utility function, an EM-based optimization framework, and two novel mechanisms: Adaptive Rejection Weighting (ARW) and Confidence-Aware Regularization (CAR). Experiments demonstrate that our approach achieves up to 9.6% higher speedup than EAGLE-3 and a 7.9% improvement in acceptance rate over ViSpec across various large language and multimodal models, significantly enhancing inference efficiency.

This paper systematically establishes an information-geometric framework for Lévy processes. **Problem:** Classical information geometry applies primarily to smooth, Gaussian, or Markovian families; extending it to non-Gaussian, non-Markovian jump processes—ubiquitous in finance—remains open. **Method:** We introduce the α-divergence for Lévy processes, from which we derive the Fisher information metric and α-connections, thereby endowing the parameter manifold with a differential-geometric structure. Our approach generalizes finite-dimensional information geometry to infinite-dimensional jump processes and yields explicit computations for tempered stable, CGMY, and variance gamma processes. **Contribution/Results:** (1) We present the first rigorous information-geometric formalism applicable to non-Gaussian, non-Markovian Lévy families; (2) we uncover intrinsic links between curvature of the parameter manifold and statistical asymptotics; and (3) we provide geometric tools for financial model selection, parameter estimation, and robustness analysis, advancing a geometric paradigm for statistical inference on stochastic processes.

This work proposes a robust, time-synchronization-free estimator for the integrated covariance matrix of multivariate continuous Itô semimartingales observed asynchronously at high frequency under market microstructure noise. The method combines local averaging with a Hayashi–Yoshida-type estimator and employs a blocking technique to circumvent conventional synchronization schemes such as previous-tick or refreshing times. Theoretical analysis establishes a central limit theorem for the proposed estimator and provides a feasible asymptotic inference framework, confirming its asymptotic normality. Simulation studies demonstrate superior finite-sample performance, highlighting its practical effectiveness in realistic settings with noisy, irregularly spaced observations.

This work aims to derive tighter information-theoretic generalization error bounds to deepen the understanding of the generalization capability of randomized learning algorithms. By establishing a unified measure-change framework grounded in the data processing inequality for f-divergences, we propose a class of general and concise inequalities that flexibly adapt to diverse settings, including conditional mutual information, PAC-Bayes, and differential privacy. This approach not only simplifies and recovers several existing state-of-the-art results but also yields novel high-probability generalization bounds across multiple learning frameworks, significantly improving both the tightness and theoretical applicability of these bounds.

Overnight rates exhibit deterministic jumps at prespecified dates—such as central bank meeting days—posing challenges for conventional short-rate models. Method: We propose the first extension of the CIR process featuring deterministic jump times, incorporating a state-dependent jump mechanism that permits both upward and downward jumps while preserving strict non-negativity and affine structure. We establish necessary and sufficient conditions for affinity, construct the process via a deterministic càdlàg time change, and extend the characterization of infinite divisibility within the affine framework. Contribution/Results: We prove existence, provide a fully calibratable explicit example, and achieve precise empirical fit to market-observed fixed-time rate jumps. The model bridges theoretical rigor—guaranteeing affinity, non-negativity, and well-posedness—with practical relevance, offering a novel paradigm for short-end interest rate modeling that accommodates institutional calendar effects.