This paper presents a non-technical account of the developments in tree-based methods for the analysis of survival data with censoring. This review describes the initial developments, which mainly extended the existing basic tree methodologies to censored data as well as to more recent work. We also cover more complex models, more specialized methods, and more specific problems such as multivariate data, the use of time-varying covariates, discrete-scale survival data, and ensemble methods applied to survival trees. A data example is used to illustrate some methods that are implemented in R.

Pricing European-style Asian options based on the arithmetic average, under the Black and Scholes model, involves estimating an integral (a mathematical expectation) for which no easily computable analytical solution is available. Pricing their American-style counterparts, which provide early exercise opportunities, poses the additional difficulty of solving a dynamic optimization problem to determine the optimal exercise strategy. A procedure for pricing American-style Asian options of the Bermudan flavor, based on dynamic programming combined with finite-element piecewise-polynomial approximation of the value function, is developed here. A convergence proof is provided. Numerical experiments illustrate the consistency and efficiency of the PROCEDURE. Theoretical properties of the value function and of the optimal exercise strategy are also established.Option Pricing, Asian Options, Path-Dependent Options, American Options, Bermudan Options, Dynamic Programming, Piecewise Polynomials

Tree‐based methods are frequently used in studies with censored survival time. Their structure and ease of interpretability make them useful to identify prognostic factors and to predict conditional survival probabilities given an individual's covariates. The existing methods are tailor‐made to deal with a survival time variable that is measured continuously. However, survival variables measured on a discrete scale are often encountered in practice. The authors propose a new tree construction method specifically adapted to such discrete‐time survival variables. The splitting procedure can be seen as an extension, to the case of right‐censored data, of the entropy criterion for a categorical outcome. The selection of the final tree is made through a pruning algorithm combined with a bootstrap correction. The authors also present a simple way of potentially improving the predictive performance of a single tree through bagging. A simulation study shows that single trees and bagged‐trees perform well compared to a parametric model. A real data example investigating the usefulness of personality dimensions in predicting early onset of cigarette smoking is presented. The Canadian Journal of Statistics 37: 17‐32; 2009 © 2009 Statistical Society of Canada

We propose a dynamic programming (DP) approach for pricing options embedded in bonds, the focus being on call and put options with advance notice. An efficient procedure is developed for the cases where the interest-rate process follows the Vasicek, Cox-IngersollRoss (CIR), or generalized Vasicek models. Our DP methodology uses the exact joint distribution of the interest rate and integrated interest rate at a future date, conditional on the current value of the interest rate. We provide numerical illustrations, for the Vasicek and CIR models, comparing our DP method with finite-difference methods. Our procedure compares quite favorably in terms of both efficiency and accuracy. An important advantage of the our DP approach is that it can be applied to more general models calibrated to capture the term structure of interest rates (e.g., the generalized Vasicek model).