An optimization approach to adaptive multi-dimensional capital management

Firms should keep capital to offer sufficient protection against the risks they are facing. In the insurance context methods have been developed to determine the minimum capital level required, but less so in the context of firms with multiple business lines including allocation. The individual capital reserve of each line can be represented by means of classical models, such as the conventional Cram\'{e}r-Lundberg model, but the challenge lies in soundly modelling the correlations between the business lines. We propose a simple yet versatile approach that allows for dependence by introducing a common environmental factor. We present a novel Bayesian approach to calibrate the latent environmental state distribution based on observations concerning the claim processes. The calibration approach is adjusted for an environmental factor that changes over time. The convergence of the calibration procedure towards the true environmental state is deduced. We then point out how to determine the optimal initial capital of the different business lines under specific constraints on the ruin probability of subsets of business lines. Upon combining the above findings, we have developed an easy-to-implement approach to capital risk management in a multi-dimensional insurance risk model

UvA-DARE (Digital Academic Repository) On the solution of Stein's equation and Fisher information matrix of an ARMAX process On the solution of Stein's equation and Fisher information matrix of an ARMAX process On the solution of Stein's equation and Fish

Abstract The main goal of this paper consists in expressing the solution of a Stein equation in terms of the Fisher information matrix (FIM) of a scalar ARMAX process. A condition for expressing the FIM in terms of a solution to a Stein equation is also set forth. Such interconnections can be derived when a companion matrix with eigenvalues equal to the roots of an appropriate polynomial associated with the ARMAX process is inserted in the Stein equation. The case of algebraic multiplicity greater than or equal to one is studied. The FIM and the corresponding solution to Stein's equation are presented as solutions to systems of linear equations. The interconnections are obtained by using the common particular solution of these systems. The kernels of the structured coefficient matrices are described as well as some right inverses. This enables us to find a solution to the newly obtained linear system of equations

Information Processes in Filtered Experiments

In this paper we give explicit representations for Kullback-Leibler information numbers between a priori and a posteriori distributions, when the observations come from a semimartingale. We assume that the distribution of the observed semimartingale is described in terms of the so-called triplet of predictable characteristics. We end by considering the corresponding notions in a model with a fractional noise