Volume 31, Issue 3

Multilevel Monte Carlo (MLMC) reduces the total computational cost of financial option pricing by combining SDE approximations with multiple resolutions. This paper explores a further avenue for reducing cost and improving power efficiency through the use of low precision calculations on configurable hardware devices such as Field-Programmable Gate Arrays (FPGAs). We propose a new framework that exploits approximate random variables and fixed-point operations with optimised precision to generate most SDE paths with a lower cost and reduce the overall cost of the MLMC framework. We first discuss several methods for the cheap generation of approximate random Normal increments. To set the bit-width of variables in the path generation we then propose a rounding error model and optimise the precision of all variables on each MLMC level. With these key improvements, our proposed framework offers higher computational savings than the existing mixed-precision MLMC frameworks.

In this study, we explore the applicability of the Feynman–Kac (FK) path integral formula to space-time fractional Schrödinger equations. In this work, a FK method based on the Lévy measure has been proposed for solving the Cauchy problems associated with the space-time fractional Schrödinger equations arising in interacting quantum systems. Application of an uncoupled Continuous Time Random Walk (CTRW) model with an exponentially distributed waiting time makes the underlying stochastic process a Lévy process which is basically a generalized Wiener process. Since these processes are Markovian in nature we can adopt the classical FK approach to simulate the CTRW model for solving the space-time fractional diffusion process with comparable simplicity and convergence rate as in the case of standard diffusion processes. Our findings underscore the viability of employing the Fractional Feynman–Kac path integral technique as an effective numerical method for solving space-time diffusion equations, thereby offering a promising alternative to traditional fractional calculus approaches.

This study deals with randomized algorithms and random projection methods for solving systems of linear algebraic equations with Toeplitz matrices. A preconditioning of such systems with circulant matrices is used that improves the convergence of the stochastic projection method. The developed stochastic algorithms are applied to first kind boundary integral equations for the Laplace, screened Poisson, and Helmholtz equations. Another application concerns the inverse problem for a wave equation where the task is to recover the unknown coefficient of this equation. A series of computer simulations are carried out to analyze the efficiency of the developed algorithm.

One way to estimate conditional expectations based on a simulation sample is to fit a parametric model to the data set and compute the conditional expectations based on the fitted model. This method may be enhanced by the control variates method. The choice of controls is critical to the effectiveness of the method. We propose the use of vanilla options as controls for estimating conditional expectations of derivative payoffs, and we provide a theoretical analysis on the uniform approximation property of vanilla options for estimating conditional expectations. Our theory suggests that vanilla options can reduce the variance of a target payoff to an arbitrarily small level subject to some assumptions. We provide examples to illustrate our proposed approach and also discuss several considerations when applying vanilla options as controls for estimating conditional expectations.

In this paper, a stochastic model related to the Rayleigh density function curve is proposed. First, we determined the explicit form of the process by solving the stochastic differential equation by applying the Itô method. Then we determined the probabilistic characteristics such as the density function, the mean and the conditional mean functions. Unlike other processes in the same context, this one allowed us to find the explicit form of the estimators of these parameters by solving the maximum likelihood equations system. In addition, an estimation study on simulated data is carried out in order to validate the efficiency of the estimators proposed by the maximum likelihood methodology. Finally, an application to real data is presented.

We show that we can generate nonrecursive 𝑛-bit pseudorandom numbers using a simple algorithm whose essential computation is five times repetition of (𝑛-bit) × \times (𝑛-bit) multiplication and taking out an 𝑛-bit integer from the result of multiplication. The algorithm can be described by 𝛽-transformation T β ⁢ ( X ) = β ⁢ X − ⌊ β ⁢ X ⌋ + 1 , X ∈ [ 1 , 2 ) , β > 1 . T_{\beta}(X)=\beta X-\lfloor\beta X\rfloor+1,\quad X\in[1,2),\,\beta>1. We consider the condition that repetition of 𝛽-transformation generates random numbers, and see why our simple algorithm works well for various values of 𝑛