Optimal probabilities and controls for reflecting diffusion processes

A solution to the optimal problem for determining vector fields which maximize (resp. minimize) the transition probabilities from one location to another for a class of reflecting diffusion processes is obtained in the present paper. The approach is based on a representation for the transition probability density functions. The optimal transition probabilities under the constraint that the drift vector field is bounded by a constant are studied in terms of the HJB equation. In dimension one, the optimal reflecting diffusion processes and the bang-bang diffusion processes are considered. We demonstrate by simulations that, even in this special case, the optimal diffusion processes exhibit an interesting feature of phase transitions. We also solve an optimal stochastic control problem for a class of stochastic control problems involving diffusion processes with reflection.Comment: 20 Pages, 2 figure

Deep Generative Modeling with Backward Stochastic Differential Equations

This paper proposes a novel deep generative model, called BSDE-Gen, which combines the flexibility of backward stochastic differential equations (BSDEs) with the power of deep neural networks for generating high-dimensional complex target data, particularly in the field of image generation. The incorporation of stochasticity and uncertainty in the generative modeling process makes BSDE-Gen an effective and natural approach for generating high-dimensional data. The paper provides a theoretical framework for BSDE-Gen, describes its model architecture, presents the maximum mean discrepancy (MMD) loss function used for training, and reports experimental results.Comment: 17 pages, 5 figure

It Ain't That Bad: Understanding the Mysterious Performance Drop in OOD Generalization for Generative Transformer Models

Generative Transformer-based models have achieved remarkable proficiency on solving diverse problems. However, their generalization ability is not fully understood and not always satisfying. Researchers take basic mathematical tasks like n-digit addition or multiplication as important perspectives for investigating their generalization behaviors. Curiously, it is observed that when training on n-digit operations (e.g., additions) in which both input operands are n-digit in length, models generalize successfully on unseen n-digit inputs (in-distribution (ID) generalization), but fail miserably and mysteriously on longer, unseen cases (out-of-distribution (OOD) generalization). Studies try to bridge this gap with workarounds such as modifying position embedding, fine-tuning, and priming with more extensive or instructive data. However, without addressing the essential mechanism, there is hardly any guarantee regarding the robustness of these solutions. We bring this unexplained performance drop into attention and ask whether it is purely from random errors. Here we turn to the mechanistic line of research which has notable successes in model interpretability. We discover that the strong ID generalization stems from structured representations, while behind the unsatisfying OOD performance, the models still exhibit clear learned algebraic structures. Specifically, these models map unseen OOD inputs to outputs with equivalence relations in the ID domain. These highlight the potential of the models to carry useful information for improved generalization

Explicit solutions for a class of nonlinear backward stochastic differential equations and their nodal sets

In this paper, we investigate a class of nonlinear backward stochastic differential equations (BSDEs) arising from financial economics, and give specific information about the nodal sets of the related solutions. As applications, we are able to obtain the explicit solutions to an interesting class of nonlinear BSDEs including the k-ignorance BSDE arising from the modeling of ambiguity of asset pricing

Large Language Models at Work in China's Labor Market

This paper explores the potential impacts of large language models (LLMs) on the Chinese labor market. We analyze occupational exposure to LLM capabilities by incorporating human expertise and LLM classifications, following Eloundou et al. (2023)'s methodology. We then aggregate occupation exposure to the industry level to obtain industry exposure scores. The results indicate a positive correlation between occupation exposure and wage levels/experience premiums, suggesting higher-paying and experience-intensive jobs may face greater displacement risks from LLM-powered software. The industry exposure scores align with expert assessments and economic intuitions. We also develop an economic growth model incorporating industry exposure to quantify the productivity-employment trade-off from AI adoption. Overall, this study provides an analytical basis for understanding the labor market impacts of increasingly capable AI systems in China. Key innovations include the occupation-level exposure analysis, industry aggregation approach, and economic modeling incorporating AI adoption and labor market effects. The findings will inform policymakers and businesses on strategies for maximizing the benefits of AI while mitigating adverse disruption risks

Enhance Diamond Coating Adhesion by Oriented Interlayer Microcracking

In this paper, we report a microcrack toughening mechanism for enhancing the adhesion of diamondcoating. The oriented microcracks were formed within the TiC interlayer to dissipate strain energy and accommodate deformation via the crack opening-closing mechanism, thus enhancing the coating/substrate interfacial toughness. The delamination of diamondcoating was effectively prevented when the parallel microcracks were confined within the interlayer and arrested at interfaces of coating/interlayer/substrate. Density functional theory calculations revealed that the highly anisotropicfracture strength of the TiC phase energetically favors crack initiation and propagation along (100) planes only, which are 54.7° away from the interface. These microcracks are constrained inside the interlayer by the two strong interfaces in the substrate/interlayer/coating system. The new microcrack toughening mechanism with these combined features has a wide application to enhance the adhesion of thin-film coatings