Volume 21, Issue 2

In this paper, we formulate and study an optimal switching problem under partial information. In our model, the agent/manager/investor attempts to maximize the expected reward by switching between different states/investments. However, he is not fully aware of his environment and only an observation process, which contains partial information about the environment/underlying, is accessible. It is based on the partial information carried by this observation process that all decisions must be made. We propose a probabilistic numerical algorithm, based on dynamic programming, regression Monte Carlo methods, and stochastic filtering theory, to compute the value function. In this paper, the approximation of the value function and the corresponding convergence result are obtained when the underlying and observation processes satisfy the linear Kalman–Bucy setting. A numerical example is included to show some specific features of partial information.

A class of stochastic particle models for the spatially discretized time-dependent Schrödinger equation is constructed. Each particle is characterized by a complex-valued weight and a position. The particle weights change according to some deterministic rules between the jumps. The jumps are determined by the creation of offspring. The main result is that certain functionals of the particle systems satisfy the Schrödinger equation. The proofs are based on the theory of piecewise deterministic Markov processes.

This paper investigates the exit-time for a broad class of symmetric finite-range jump processes via the corresponding master equation, a nonlocal diffusion equation suitably constrained. In direct analogy to the classical diffusion equation with a homogeneous Dirichlet boundary condition, the nonlocal diffusion equation is augmented with a homogeneous volume-constraint. The volume-constrained master equation provides an efficient alternative over Monte Carlo simulation for computing an important statistic of the process. Several numerical examples are given.

The paper deals with a stochastic analysis of random displacements governed by isotropic elasticity equations. The elasticity constants are assumed to be random fields with gaussian distribution. Under the assumption of small fluctuations of elasticity constants but not ignoring their distribution and correlation structure, we derive the spectral tensor of the random displacement field. A randomized spectral representation is then used to simulate this random field numerically. The same approach was applied to the case of random loading. In this case, the intensity of fluctuations may be arbitrarily large. A series of test calculations confirm the high accuracy and computational efficiency of the method.

Graphics Processing Units (GPUs) bring the promise of supercomputing power for a fraction of the cost of traditional supercomputing, with possible speed-ups over comparable CPU hardware of one or two orders of magnitude. Rapid development of both proprietary libraries, such as NVIDIA's CUDA, and an open standard, OpenCL, have opened the doors to the GPU's cheap computing power. Unfortunately, random number generators (RNGs) have been slow to catch up with the rapid expansion of GPU computing. The number of types of RNGs available for GPUs is small, and the statistical quality of those provided with standard libraries are frequently unknown. Because specific RNGs may have statistical quality for certain applications, new kinds of RNGs must be made available for GPU computing to bring the full power of GPUs to different kinds of research. Lagged-Fibonacci Generators (LFGs), in particular, have been difficult to develop for memory-challenged GPUs because of their large state space, which is unfortunate because they have excellent statistical properties for many applications. In this paper, we discuss our implementation of memory efficient, integer, cycle-split, additive and multiplicative LFGs for both CUDA and OpenCL. The latter LFG has been implemented neither for GPUs nor as a split-steam parallel generator before. We also discuss portability and reproducibility between CPUs and GPUs.

A family of d -dimensional functions is introduced. It includes the test functions that in [Monte Carlo Methods Appl. 20 (2014), no. 3, 167–171] demonstrate the efficiency of certain quasi-Monte Carlo integrations. The average dimensions of these d -dimensional functions converge as d → ∞.