In this article, we calibrate the Vasicek interest rate model under the risk neutral measure by learning the model parameters using Gaussian processes for machine learning regression. The calibration is done by maximizing the likelihood of zero coupon bond log prices, using mean and covariance functions computed analytically, as well as likelihood derivatives with respect to the parameters. The maximization method used is the conjugate gradients. The only prices needed for calibration are zero coupon bond prices and the parameters are directly obtained in the arbitrage free risk neutral measure.Keywords Arbitrage free risk neutral measure; Calibration; Gaussian processes for machine learning; Vasicek interest rate model; Zero coupon bond prices.
We present a methodology to connect an ordinary differential equation (ODE) model of interacting entities at the individual level, to an open Markov chain (OMC) model of a population of such individuals, via a stochastic differential equation (SDE) intermediate model. The ODE model here presented is formulated as a dynamic change between two regimes; one regime is of mean reverting type and the other is of inverse logistic type. For the general purpose of defining an OMC model for a population of individuals, we associate an Ito processes, in the form of SDE to ODE system of equations, by means of the addition of Gaussian noise terms which may be thought to model non essential characteristics of the phenomena with small and undifferentiated influences. The next step consists on discretizing the SDE and using the discretized trajectories computed by simulation to define transitions of a finite valued Markov chain; for that, the state space of the Ito processes is partitioned according to some rule. For the example proposed for illustration, the state space of the ODE system referred – corresponding to a model of a viral infection – is partitioned into six infection classes determined by some of the critical points of the ODE system; we detail the evolution of some infected population in these infection classes.
We consider a non-homogeneous continuous time Markov chain model for Long-Term Care with five states: the autonomous state, three dependent states of light, moderate and severe dependence levels and the death state. For a general approach, we allow for non null intensities for all the returns from higher dependence levels to all lesser dependencies in the multi-state model. Using data from the 2015 Portuguese National Network of Continuous Care database, as the main research contribution of this paper, we propose a method to calibrate transition intensities with the one step transition probabilities estimated from data. This allows us to use non-homogeneous continuous time Markov chains for modeling Long-Term Care. We solve numerically the Kolmogorov forward differential equations in order to obtain continuous time transition probabilities. We assess the quality of the calibration using the Portuguese life expectancies. Based on reasonable monthly costs for each dependence state we compute, by Monte Carlo simulation, trajectories of the Markov chain process and derive relevant information for model validation and premium calculation.
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