  In the last five years, the financial industry has been impacted by the
emergence of digitalization and machine learning. In this article, we explore
two methods that have undergone rapid development in recent years: Gaussian
processes and Bayesian optimization. Gaussian processes can be seen as a
generalization of Gaussian random vectors and are associated with the
development of kernel methods. Bayesian optimization is an approach for
performing derivative-free global optimization in a small dimension, and uses
Gaussian processes to locate the global maximum of a black-box function. The
first part of the article reviews these two tools and shows how they are
connected. In particular, we focus on the Gaussian process regression, which is
the core of Bayesian machine learning, and the issue of hyperparameter
selection. The second part is dedicated to two financial applications. We first
consider the modeling of the term structure of interest rates. More precisely,
we test the fitting method and compare the GP prediction and the random walk
model. The second application is the construction of trend-following
strategies, in particular the online estimation of trend and covariance
windows.
