Mathematical structures in economics and finance

传统上，数学的应用一直在自然科学中寻求，我们学科的大部分历史进步来自于回答物理学，工程学以及最近的生物学提出的问题。我一直相信，社会科学与自然科学具有同样的潜力，真正理解经济学所提出的问题，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（和数学家的新职位）。另一方面，老问题不应该被忽视，因为（a）它们作为测试新数学方法的基准，（例如，近年来辛几何的巨大进步，这在很大程度上是由于非线性分析的进步），以及（B）数学中存在深刻的统一，在一个领域中获得的结果可能在另一个领域中有用（例如，随机分析，包括其最新发展，如Malliavin微积分，对现代金融产生了深远的影响）。

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).