PDE

There is a beautiful connection between probability theory and partial differential equations (PDEs), given by the famous Feynman–Kac theorem. This result allows us to translate between finite-dimensional stochastic problems and infinite-dimensional deterministic problems. It appears throughout finance, physics, control theory, and machine learning (generative modeling and reinforcement learning). What makes this connection especially useful is that it works in both directions. The solution to a high-dimensional PDE (which can be very expensive to compute numerically) can be evaluated at a single point simply by simulating a random process. Conversely, a difficult question about a stochastic process can be transformed into a deterministic PDE and tackled using standard techniques. ...

This post is an extract of my Master’s Thesis in Mathematics, it can be found in this GitHub repository for a complete and autocomplete document: repo. In our continuous pursuit of enlightenment, knowledge, and understanding of the world around us, we turn to the tool of mathematics. For thousands of years, mathematicians have sought to translate our sensory experiences into a language that captures their essence in the most abstract, universal, and general form. ...