Understanding rare events for efficient algorithms

Federica Milinanni will receive her doctoral degree in mathematics from KTH Royal Institute of Technology in 2025. Thanks to a grant from the Knut and Alice Wallenberg Foundation, she will hold a postdoctoral position with Professor Chang-Han Rhee, Northwestern University, Evanston, USA.

Probability theory is a mathematical tool for analysing random events. To determine the probability that random events occur, we usually use mathematical models that involve probability distributions. A common example is the normal distribution, also known as the Gaussian distribution.

It is possible, with a very small probability, that an outcome deviates significantly from the expected behaviour. For example, when flipping a coin many times, we expect heads to appear roughly half of the times. However, in rare cases, heads may appear in the majority of flips. This kind of rare result is addressed by the theory of large deviations. Studying the probability that rare events occur is extremely important in many fields and, in the late 1930s, Swedish insurance mathematician Harald Cramér was the first to study this. This theory, in its modern form, is central to the project.

To calculate probabilities of random events, we can rely on numerical methods. The most common algorithms are called Monte Carlo methods; these build upon repeatedly simulating a large amount of artificial random data and then computing the average result.

Simulations are often extremely time consuming for large data sets or complex systems. Therefore, the aim of the project is to study and design more efficient algorithms for calculating probabilities using the theory of large deviations. This will allow Monte Carlo methods to become both faster and more reliable, so they can better address the challenges of real-world applications, mainly in insurance, finance, and computational biology.