The aim of the paper is to derive a simple, implementable machine learning method for general insurance losses. An algorithm for learning a general insurance loss triangle is developed and justified. An argument is made for applying support vector regression (SVR) to this learning task (in order to facilitate transparency of the learning method as compared to more “black-box” methods such as deep neural networks), and SVR methodology derived is specifically applied to this learning task. A further argument for preserving the statistical features of the loss data in the SVR machine is made. A bespoke kernel function that preserves the statistical features of the loss data is derived from first principles and called the exponential dispersion family (EDF) kernel. Features of the EDF kernel are explored, and the kernel is applied to an insurance loss estimation exercise for homogeneous risk of three different insurers. Results of the cumulative losses and ultimate losses predicted by the EDF kernel are compared to losses predicted by the radial basis function kernel and the chain-ladder method. A backtest of the developed method is performed. A discussion of the results and their implications follows.

In this paper we discuss the range of a co-analytic Toeplitz operator. These range spaces are closely related to de Branges–Rovnyak spaces (in some cases they are equal as sets). In order to understand its structure, we explore when the range space decomposes into the range of an associated analytic Toeplitz operator and an identifiable orthogonal complement. For certain cases, we compute this orthogonal complement in terms of the kernel of a certain Toeplitz operator on the Hardy space, where we focus on when this kernel is a model space (backward shift invariant subspace). In the spirit of Ahern–Clark, we also discuss the non-tangential boundary behavior in these range spaces. These results give us further insight into the description of the range of a co-analytic Toeplitz operator as well as its orthogonal decomposition. Our Ahern–Clark type results, which are stated in a general abstract setting, will also have applications to related sub-Hardy Hilbert spaces of analytic functions such as the de Branges–Rovnyak spaces and the harmonically weighted Dirichlet spaces.

We propose a new primal-dual interior-point algorithm based on a new kernel function for linear optimization problems. New search directions and proximity functions are proposed based on the kernel function. We show that the new algorithm has and iteration bounds for large-update and small-update methods, respectively, which are currently the best known bounds for such methods.

Interior-Point Methods (IPMs) are not only very effective in practice for solving linear optimization problems but also have polynomial-time complexity. Despite the practical efficiency of large-update algorithms, from a theoretical point of view, these algorithms have a weaker iteration bound with respect to small-update algorithms. In fact, there is a significant gap between theory and practice for large-update algorithms. By introducing self-regular barrier functions, Peng, Roos and Terlaky improved this gap up to a factor of log n. However, checking these self-regular functions is not simple and proofs of theorems involving these functions are very complicated. Roos el al. by presenting a new class of barrier functions which are not necessarily self-regular, achieved very good results through some much simpler theorems. In this paper we introduce a new kernel function in this class which yields the best known complexity bound, both for large-update and small-update methods.

We study path properties of kernel generated two-time parameter, not necessarily stationary, Gaussian processes. We establish large deviation results for some increments of these processes and use these results to prove some of their moduli of continuity and other path properties.