Overcoming the curse of dimensionality: stochastic algorithms for high-dimensional partial differential equations (CURSEOFDIM)

Basic data for this project

Description

Partial differential equations (PDEs) are among the most universal tools used in modeling problems in nature and man-made complex systems. In particular, PDEs are a fundamental tool in portfolio optimization problems and in the state-of-the-art pricing and hedging of financial derivatives. The PDEs appearing in such financial engineering applications are often high dimensional as the dimensionality of the PDE corresponds to the number of financial asserts in the involved hedging portfolio. Such PDEs can typically not be solved explicitly anddeveloping efficient numerical algorithms for high dimensional PDEs is one of the most challenging tasks in applied mathematics. As is well-known, the difficulty lies in the so-called ``curse of dimensionality'' in the sense that the computational effort of standard approximation algorithms grows exponentially in the dimension of the considered PDE and there is only a very limited number of cases where a practical PDE approximation algorithm with a computational effort which grows at most polynomially in the PDE dimension has been developed. It is the key objective of this research project to overcome this curse of dimensionality and to construct and analyze new approximation algorithms which solve high dimensional PDEs with a computational effffort that grows at most polynomially in both the dimension of the PDE and the reciprocal of the prescribed approximation precision. Key tools, which we intend to use to achieve this objective, are suitable nonlinear Feynman-Kac formulas, stochastic approximation algorithms, and deep neural networks.