This review explores the role of prompt engineering in unleashing the capabilities of large language models (LLMs). Prompt engineering is the process of structuring inputs, and it has emerged as a crucial technique for maximizing the utility and accuracy of these models. Both foundational and advanced prompt engineering methodologies—including techniques such as self-consistency, chain of thought, and generated knowledge, which can significantly enhance the performance of models—are explored in this paper. Additionally, the prompt methods for vision language models (VLMs) are examined in detail. Prompt methods are evaluated with subjective and objective metrics, ensuring a robust analysis of their efficacy. Critical to this discussion is the role of prompt engineering in artificial intelligence (AI) security, particularly in terms of defending against adversarial attacks that exploit vulnerabilities in LLMs. Strategies for minimizing these risks and improving the robustness of models are thoroughly reviewed. Finally, we provide a perspective for future research and applications.

This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first...

: We propose a probabilistic numerical algorithm to solve Backward Stochastic Differential Equations (BSDEs) with nonnegative jumps, a class of BSDEs introduced in [`Feynman–Kac representation for Hamilton–Jacobi–Bellman IPDE', Ann. Probab., to appear] for representing fully nonlinear HJB equations. This includes in particular numerical resolution for stochastic control problems with controlled volatility, possibly degenerate. Our backward scheme, based on least-squares regressions, takes advantage of high-dimensional properties of Monte Carlo methods, and also provides a parametric estimate in feedback form for the optimal control. A partial analysis of the algorithm error is presented, as well as numerical tests on the problem of option superreplication with uncertain volatilities and/or correlations, including a detailed comparison with the numerical results from the alternative scheme proposed in [J. Comput. Finance 14 (2011), 37–71].

Kernel density estimation and kernel regression are powerful but computationally expensive techniques: a direct evaluation of kernel density estimates at M evaluation points given N input sample points requires a quadratic $\mathcal{O}(MN)$ operations, which is prohibitive for large scale problems. For this reason, approximate methods such as binning with Fast Fourier Transform or the Fast Gauss Transform have been proposed to speed up kernel density estimation. Among these fast methods, the Fast Sum Updating approach is an attractive alternative, as it is an exact method and its speed is independent of the input sample and the bandwidth. Unfortunately, this method, based on data sorting, has for the most part been limited to the univariate case. In this paper, we revisit the fast sum updating approach and extend it in several ways. Our main contribution is to extend it to the general multivariate case for general input data and rectilinear evaluation grid. Other contributions include its extension to a wider class of kernels, including the triangular, cosine and Silverman kernels, its combination with parsimonious additive multivariate kernels, and its combination with a fast approximate k-nearest-neighbors bandwidth for multivariate datasets. Our numerical tests confirm the speed, accuracy and stability of the method.