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A common assumption in supervised learning is that the training and test input points follow the same probability distribution. However, this assumption is not fulfilled, e.g., in interpolation, extrapolation, active learning, or classification with imbalanced data. The violation of this assumption—known as the covariate shift —causes a heavy bias in standard generalization error estimation schemes such as cross-validation or Akaike's information criterion, and thus they result in poor model selection. In this paper, we propose an alternative estimator of the generalization error for the squared loss function when training and test distributions are different. The proposed generalization error estimator is shown to be exactly unbiased for finite samples if the learning target function is realizable and asymptotically unbiased in general. We also show that, in addition to the unbiasedness, the proposed generalization error estimator can accurately estimate the difference of the generalization error among different models, which is a desirable property in model selection. Numerical studies show that the proposed method compares favorably with existing model selection methods in regression for extrapolation and in classification with imbalanced data.

The forward rate curve is assumed to follow a stochastic differential equation w.r.t. a Lévy process with infinite dimensions. Conditions under which the market is free of arbitrage are provided for both the interest rate case and for the case of credit risk with ratings. A simulation shows that typical movements of the yield curve are well captured by the model.

The paper develops the method of quantile hedging in a two-factor jump-diffusion market. The exact formulae of the maximal successful hedging set for an option to exchange one asset for another are given. These results are applied to a class of equity-linked life insurance contracts called “pure endowments with a guarantee”. In our setting, the pay-off functions of these insurance contracts are equal to the maximum of two risky assets in a two-factor jump-diffusion model conditioned by the survival status of the insured. The first asset is responsible for the maximal size of future profits, while the second provides a flexible guarantee to the insured. Based on quantile hedging methodology and a generalized Margrabe's formula, the paper describes the valuation and risk management of such mixed finance-insurance instruments.

The Cauchy problem for the one-dimensional heat equation asks for solutions u f ( x,t ) of ∂ u /∂ t = ∂ 2 u / ∂ x 2 on R × [1, ∞) with u ( x , 1) = f ( x ) on R. Here we assume that the initial condition f ( x ), x ∊ R, and hence the solution u f is unknown but that at times t j , j = 1, 2, …, n , noisy measurements are available from which an estimator ~ f n of the initial condition may be obtained. The paper studies the asymptotics (as t → ∞ and n → ∞) of u f ( xt 1/2 , t )– u ~fn ( xt 1/2 , t ) in mean integrated squared error.