Mathematical structures in economics and finance

传统上，人们一直在自然科学中寻求数学的应用，我们学科的许多历史进步都来自于对物理学、工程学以及最近的生物学提出的问题的回答。我一直相信，社会科学与自然科学具有同样的潜力，对经济学所提出的问题的真正理解，例如，将在理论和数值层面上提出新的数学问题，并为数学开辟新的方向（并为数学家开辟新的职位）。另一方面，老问题也不应该被忽视，因为(a)它们可以作为检验新数学方法的基准（例如，近年来辛几何的巨大进步，很大程度上是由于非线性分析的进步），(b)数学具有深刻的统一性，在一个领域获得的结果可能在另一个领域有用（例如，随机分析，包括其最近的发展）。比如马利亚文微积分（Malliavin calculus），对现代金融产生了深远影响。

Applications of mathematics have traditionally been sought in the natural sciences, and much of the historical progress of our discipline comes from answering questions raised by physics, engineering and, more recently, biology. I have always been convinced that the social sciences have the same potential as the natural ones, and that a true understanding of the problems raised by economics, for instance, will raise new mathematical problems, both at the theoretical and numerical level, and open new directions for mathematics (and new positions for mathematicians). On the other hand, old problems should not be neglected, because (a) they serve as a benchmark against which to test new mathematical methods which (witness, for instance, the tremendous advances of symplectic geometry in recent years, which is largely due to the progress of nonlinear analysis), and (b) there is a deep unity in mathematics, and results obtained in one field can turn out to be useful in another (stochastic analysis, for instance, including its most recent development, such as Malliavin calculus, has deeply influenced modern finance).