狄拉克函数在金融数学进阶学习中的应用与教学实践研究

Applications and Teaching Practices of the Dirac Delta Function in Advanced Financial Mathematics Courses

摘要: 金融数学作为基础数学理论在实际问题中的典型应用，对数学专业学生具有很好的实践学习价值，可帮助学生理解抽象理论背后的丰富意义与广泛应用，增强学生的学习兴趣。在课程教学中做到既重视金融知识等业务背景的介绍，又凸出数学知识的理论高度是当前金融数学教学过程中面临的一大制约限制。本论文以狄拉克函数作为切入点，通过将其链接离散、连续的特性与金融数学中的年金计算进行类比与统一，贡献了兼顾数学理论与金融实践的典型案例，不仅后续可作为教材与教学改革中的核心元素，还为师生提供了金融数学进阶学习的方向。进一步，通过提出“课前–课中–课后”三位一体的教学实践建议，为该案例在金融数学课程的实际应用提供了良好参考。

Abstract: Financial mathematics, as a foundational mathematical theory, holds significant practical learning value for mathematics majors by illustrating the rich implications and broad applications behind abstract concepts, thereby enhancing students’ engagement and interest. A key challenge in its pedagogy lies in balancing the introduction of financial and business contexts with the rigorous theoretical depth required in mathematical education. This paper takes the Dirac delta function as a starting point, drawing analogies and unifying its discrete and continuous characteristics with annuity calculations in financial mathematics to provide a compelling case study that integrates mathematical theory and financial practice. This case can subsequently serve as a core element in curriculum design and pedagogical reforms, while also pointing to direction for advance study of Financial Mathematics. Furthermore, by proposing a three-in-one teaching framework (“pre-class - in-class - post-class”), the paper offers actionable guidance for implementing this case in financial mathematics courses, ensuring its practical effectiveness.