Reciprocal processes are stochastic processes such that the current state only depends on the nearest past and future, and they have found many applications in various fields such as Euclidean quantum physics. This thesis focuses on the study of some classes of reciprocal processes in both discrete-time and continuous-time frameworks.  

The main contributions of this thesis can be splitted into two parts. The first and major part of this thesis consists of the study of several discrete-time reciprocal processes: reciprocal chains, hidden Markov models (HMMs) and hidden reciprocal models (HRMs). More specifically, we (i) formally define reciprocal chains and explore their properties and similarities/differences to Markov chains; (ii) point out that one of the three most commonly used definitions of HMMs has fatal flaws and list some key properties of HMMs; (iii) present the connections between undirected graphical models and HMMs/reciprocal chains; (iv) learn and compare HMMs and HRMs parameters based on the expectation-maximization algorithm.  

The second part of the thesis focuses on the study of Brownian bridges with pre-scribed terminal densities which are special continuous-time reciprocal processes. We derive large deviations for the family of Brownian bridges with the help of Girsanov transformation.