Synthetic augmentation is increasingly used to mitigate data scarcity in financial machine learning, yet its statistical role remains poorly understood. We formalize synthetic augmentation as a modification of the effective training distribution and show that it induces a structural bias--variance trade-off: while additional samples may reduce estimation error, they may also shift the population objective whenever the synthetic distribution deviates from regions relevant under evaluation. To isolate informational gains from mechanical sample-size effects, we introduce a size-matched null augmentation and a finite-sample, non-parametric block permutation test that remains valid under weak temporal dependence. We evaluate this framework in both controlled Markov-switching environments and real financial datasets, including high-frequency option trade data and a daily equity panel. Across generators spanning bootstrap, copula-based models, variational autoencoders, diffusion models, and TimeGAN, we vary augmentation ratio, model capacity, task type, regime rarity, and signal-to-noise. We show that synthetic augmentation is beneficial only in variance-dominant regimes, such as persistent volatility forecasting-while it deteriorates performance in bias-dominant settings, including near-efficient directional prediction. Rare-regime targeting can improve domain-specific metrics but may conflict with unconditional permutation inference. Our results provide a structural perspective on when synthetic data improves financial learning performance and when it induces persistent distributional distortion.

Denoising diffusion probabilistic models (DDPMs) have emerged as powerful generative models for complex distributions, yet their use in arbitrage-free derivative pricing remains largely unexplored. Financial asset prices are naturally modeled by stochastic differential equations (SDEs), whose forward and reverse density evolution closely parallels the forward noising and reverse denoising structure of diffusion models. In this paper, we develop a framework for using DDPMs to generate risk-neutral asset price dynamics for derivative valuation. Starting from log-return dynamics under the physical measure, we analyze the associated forward diffusion and derive the reverse-time SDE. We show that the change of measure from the physical to the risk-neutral measure induces an additive shift in the score function, which translates into a closed-form risk-neutral epsilon shift in the DDPM reverse dynamics. This correction enforces the risk-neutral drift while preserving the learned variance and higher-order structure, yielding an explicit bridge between diffusion-based generative modeling and classical risk-neutral SDE-based pricing. We show that the resulting discounted price paths satisfy the martingale condition under the risk-neutral measure. Empirically, the method reproduces the risk-neutral terminal distribution and accurately prices both European and path-dependent derivatives, including arithmetic Asian options, under a GBM benchmark. These results demonstrate that diffusion-based generative models provide a flexible and principled approach to simulation-based derivative pricing.

We consider the problem of inferring high-dimensional datax in a model that consists of a priorp(x) and an auxiliary differentiable constraintc(x,y) on x given some additional informationy. In this paper, the prior is an independently trained denoising diffusion generative model. The auxiliary constraint is expected to have a differentiable form, but can come from diverse sources.

Their Markovian structure makes it difficult to define DDPMs with distributions other than Gaussian or discrete. In this paper, we introduce Star-Shaped DDPM (SS-DDPM).

Surprisingly, we show that even the addition ofnon-realistic random data (generated byGaussian sampling) can improverobustness.

Diffusion models have become a central tool in deep generative modeling, but standard formulations rely on a single network and a single diffusion schedule to transform a simple prior, typically a standard normal distribution, into the target data distribution. As a result, the model must simultaneously represent the global structure of the distribution and its fine-scale local variations, which becomes difficult when these scales are strongly mismatched. This issue arises both in natural images, where coarse manifold-level structure and fine textures coexist, and in low-dimensional distributions with highly concentrated local structure. To address this issue, we propose Residual Prior Diffusion (RPD), a two-stage framework in which a coarse prior model first captures the large-scale structure of the data distribution, and a diffusion model is then trained to represent the residual between the prior and the target data distribution. We formulate RPD as an explicit probabilistic model with a tractable evidence lower bound, whose optimization reduces to the familiar objectives of noise prediction or velocity prediction. We further introduce auxiliary variables that leverage information from the prior model and theoretically analyze how they reduce the difficulty of the prediction problem in RPD. Experiments on synthetic datasets with fine-grained local structure show that standard diffusion models fail to capture local details, whereas RPD accurately captures fine-scale detail while preserving the large-scale structure of the distribution. On natural image generation tasks, RPD achieved generation quality that matched or exceeded that of representative diffusion-based baselines and it maintained strong performance even with a small number of inference steps.

Train-time data poisoning attacks threaten machine learning models by introducing adversarial examples during training, leading to misclassification. Current defense methods often reduce generalization performance, are attack-specific, and impose significant training overhead. To address this, we introduce a set of universal data purification methods using a stochastic transform, $\Psi(x)$, realized via iterative Langevin dynamics of Energy-Based Models (EBMs), Denoising Diffusion Probabilistic Models (DDPMs), or both. These approaches purify poisoned data with minimal impact on classifier generalization. Our specially trained EBMs and DDPMs provide state-of-the-art defense against various attacks (including Narcissus, Bullseye Polytope, Gradient Matching) on CIFAR-10, Tiny-ImageNet, and CINIC-10, without needing attack or classifier-specific information. We discuss performance trade-offs and show that our methods remain highly effective even with poisoned or distributionally shifted generative model training data.