Dylan Possamaï

Takashi Furuya, Anastasis Kratsios, Dylan Possamaï et al.

We introduce a mild generative variant of the classical neural operator model, which leverages Kolmogorov--Arnold networks to solve infinite families of second-order backward stochastic differential equations ($2$BSDEs) on regular bounded Euclidean domains with random terminal time. Our first main result shows that the solution operator associated with a broad range of $2$BSDE families is approximable by appropriate neural operator models. We then identify a structured subclass of (infinite) families of $2$BSDEs whose neural operator approximation requires only a polynomial number of parameters in the reciprocal approximation rate, as opposed to the exponential requirement in general worst-case neural operator guarantees.

Johannes Schmitt, Gergely Bérczi, Jasper Dekoninck et al.

As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 39 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation of mathematical reasoning capabilities by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can succeed at the more accessible research-level questions, but still encounter significant difficulties on more challenging problems. Quantitatively, Grok-4 achieves the highest accuracy of 52% on final-answer subproblems, while GPT-5 obtains the best performance for proof generation, achieving a fully correct solution for 22% of problems. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.

Dylan Possamaï, Xiaolu Tan

We study the weak approximation of the second-order backward SDEs (2BSDEs), when the continuous driving martingales are approximated by discrete time martingales. We establish a convergence result for a class of 2BSDEs, using both robustness properties of BSDEs, as proved in Briand, Delyon and Mémin [Stochastic Process. Appl. 97 (2002) 229-253], and tightness of solutions to discrete time BSDEs. In particular, when the approximating martingales are given by some particular controlled Markov chains, we obtain several concrete numerical schemes for 2BSDEs, which we illustrate on specific examples.