Incomplete information is at the heart of information-based credit risk models. In this paper, we rigorously define incomplete information with the notion of "delayed filtrations". We characterize two distinct types of delayed information, continuous and discrete: the first generated by a time change of filtrations and the second by finitely many marked point processes. This notion unifies the noisy information in Duffie and Lando (2001) and the partial information in Collin-Dufresne et al. (2004), under which structural models are translated into reduced-form intensity-based models. We illustrate through a simple example the importance of this notion of delayed information, as well as the potential pitfall for abusing the Laplacian approximation techniques for calculating the intensity process in an information-based model. * The authors thank the Associate Editor and the two anonymous referees for their careful reading, constructive suggestions, and enlightening remarks.

Let (Xt) t≥0 be a continuous-time, time-homogeneous strong Markov process with possible jumps and let τ be its first hitting time of a Borel subset of the state space. Suppose X is sampled at random times and suppose also that X has not hit the Borel set by time t. What is the intensity process of τ based on this information?This question from credit risk encompasses basic mathematical problems concerning the existence of an intensity process and filtration expansions, as well as some conceptual issues for credit risk. By revisiting and extending the famous Jeulin-Yor [Lecture Notes in Math. 649 (1978) 78-97] result regarding compensators under a general filtration expansion framework, a novel computation methodology for the intensity process of a stopping time is proposed. En route, an analogous characterization result for martingales of Jacod and Skorohod [Lecture Notes in Math. 1583 (1994) 21-35] under local jumping filtration is derived. . This reprint differs from the original in pagination and typographic detail. 1 2 X. GUO AND Y. ZENGrisk, especially in the information-based approach pioneered by Duffie and Lando [14]. First, if τ is the default time of a firm and is a stopping time relative to some filtration G = (G t ) t≥0 , then under appropriate technical conditions (such as those in Aven [1]), λ t is the instantaneous likelihood of default at time t conditioned on G t , the information at time t. That is,

Crown volume is an important tree factor used in forest surveys as a prerequisite for estimating biomass and carbon stocks. This study developed a method for accurately calculating the crown volume of individual trees from vehicle-borne laser scanning (VLS) data using a concave hull by slices method. CloudCompare, an open-source three-dimensional (3D) point cloud and mesh processing software package, was used with VLS data to segment individual trees from which single tree crowns were extracted by identifying the first branch point of the tree. The slice thickness and number to be fitted to the canopy point cloud were adaptively determined based on the change rate in area with height, with the area of each slice calculated using the concave hull algorithm with portions of the crown regarded as truncated cones. The overall volume was then calculated as the sum of all sub-volumes. The proposed method was experimentally validated on 30 urban trees by comparing the crown volumes calculated using the proposed method with those calculated using five existing methods (manual measurement, 3D convex hull, 3D alpha shape, convex hull by slices, and voxel-based). The proposed method produced the smallest average crown volume. Gaps and holes in the point cloud were regarded as part of the crown by the manual measurement, 3D convex hull, and convex hull by slices method, resulting in the calculated volume being higher than the true value; the proposed method reduced this effect. These results indicate that the concave hull by slices method can more effectively calculate the crown volume of a single tree from VLS data.

This paper provides a model for the recovery rate process in a reduced form model. After default, a firm continues to operate, and the recovery rate is determined by the value of the firm's assets relative to its liabilities. The debt recovers a different magnitude depending upon whether or not the firm enters insolvency and bankruptcy. Although this recovery rate process is similar to that used in a structural model, the reduced form approach is maintained by utilizing information reduction in the sense of Guo, Jarrow, and Zeng. Our model is able to provide analytic expressions for a firm's default intensity, bankruptcy intensity, and zero-coupon bond prices both before and after default.

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