Sobol' sequence generators are used actively in financial applications. In this paper, we explore the effect of the uniformity Properties A and A' on the generator performance in high-dimensional problems. It is shown that these properties provide an additional guarantee of uniformity for highdimensional problems even at a small number of sampled points. By imposing additional uniformity properties on low-dimensional projections of the sequence in addition to the uniformity properties of the d-dimensional sequence itself, the efficiency of the Sobol' sequence can be increased. The SobolSeq16384 generator, which satisfies additional uniformity properties (Property A for all 16,384 dimensions and Property A' for adjacent dimensions), is constructed. A comparison of known Sobol' sequence generators for a set of tests shows that for the majority of tests, the SobolSeq16384 generator performs better than other generators.

In this paper we consider the following inverse problem for the first hitting time distribution: given a Wiener process with a random initial state, probability distribution, F (t), and a linear boundary, b(t) = µt, find a distribution of the initial state such that the distribution of the first hitting time is F (t). This problem has important applications in credit risk modeling where the process represents, so-called, distance to default of an obligor, the first hitting time represents a default event and the boundary separates the healthy states of the obligor from the default state. We show that randomization of the initial state of the process makes the problem analytically tractable.Primary Subjects: 60G40 Secondary Subjects: 91B70

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