Feynman–Kac formula
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 stochastic finite-dimensional problems and deterministic infinite-dimensional problems. It appears everywhere in 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. A high-dimensional PDE (very expensive to solve 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. ...