In this paper, we propose a new bilinear Log-GARCH model. We establish a sufficient condition ensuring that the proposed model admits a unique, strictly stationary, and ergodic solution. We then apply the quasi-maximum likelihood estimation (QMLE) method to estimate the model parameters. Several theorems are presented to demonstrate the convergence of the estimation procedure and the convergence of the estimated parameters to the true values. Furthermore, we implement a Monte Carlo simulation to generate the time series. The purpose of these simulations is to assess the ability of the proposed model to capture the dynamics of financial volatility, particularly asymmetry and leverage effects, as well as to evaluate the performance of the QMLE estimator in finite samples. Additionally, we present carefully selected numerical examples that produce remarkable results, confirming both the effectiveness of the bilinear log-GARCH model and the reliability of the Monte Carlo simulation framework, as well as the accuracy of the parameter estimation. These results demonstrate that the proposed model is capable of capturing the complex volatility behavior in financial time series, making it a suitable tool for analyzing financial data with seasonal or nonlinear characteristics.
This paper considers a single server queue subject to catastrophes with service lateness which consists on control and service, in batch of fixed-size. Catastrophes can occur when the server is on service, or on vacation. Whenever a catastrophe occurs, all customers present in the system are destroyed immediately and the server goes for multiple vacations until it finds at least a batch of K customers in the queue, at which point the control period of this batch begins in the service lateness. After completion of the control time, this batch enters the service immediately. This proposed model is solved by giving explicit formulas for the steady-state probabilities and some performance measures. Numerical results are sketched out to illustrate the effect of the system parameters on the main performance measures. In addition, a Monte Carlo simulation is performed using three sampling methods to manage the proposed model with large batches. We used the generators "rand", "goLHS" and "goRDS" to sample the inputs for the simulation. The results have shown that RDS improves significantly the studied performance measures of the queue compared to the Monte Carlo method and LHS by considering the variance reduction and the relative error as comparison criteria.
The well-known Random Walk on Spheres method (RWS) for the Laplace equation is here extended to drift-diffusion-reaction problems with spatially varying velocity and reaction rate coefficients. The RWS algorithm is based on a generalized spherical mean value relation with constant velocity and reaction rate which is an extension of the classical spherical mean value relation for the Laplace equation. Heuristically, the same algorithm can be applied when the velocity and the reaction rate coefficients are spatially varying just by choosing small spheres in the RWS process assuming the coefficients in small spheres are approximately constant. A fundamental question concerns the quantitative assessment of the error induced by this approximation. In the present work, we establish a rigorous bound for this error. The paper also presents the results of numerical experiments performed to validate the obtained error estimate.
We propose soft-random numbers (SRNs), a novel class of random sequences designed to interpolate between quasi-random and pseudo-random regimes through a tunable softening parameter in [ 0 , 1 ] [0,1] . Values near zero yield low-discrepancy quasi-random numbers, while values close to one generate pseudo-random sequences, thus enabling flexible control over discrepancy. The discrepancy behavior of SRNs is analyzed across multiple sample sizes and distributions, with statistical validity assessed using two goodness-of-fit tests. Comparative evaluations demonstrate that SRNs achieve competitive low-discrepancy properties while maintaining computational efficiency, outperforming or complementing established methods such as scrambled Sobol sequences and jittered sampling. Applications in Monte Carlo integration, uncertainty quantification, and stochastic decision processes illustrate the practical relevance of SRNs. Overall, the proposed approach provides a computationally efficient and flexible framework for random number generation, bridging the gap between deterministic quasi-random and stochastic pseudo-random techniques.
A general framework for solving stochastic boundary value problems (SBVPs) in diffusion-type stochastic differential equations (SDEs) is developed based on the decomposition of the transition probability density function. The proposed approach relies on the Hermite polynomial expansion of the conditional transition density and its spatial derivatives, which allows for a direct construction of guided stochastic bridges connecting prescribed boundary points. The method unifies simulation, analytical approximation, and statistical reweighting in a single probabilistic representation that remains valid for nonlinear, non-Gaussian, and anisotropic diffusion systems. Several algorithms for guided bridge simulation are formulated and compared, including Gaussian-only approximation, Gaussian-Hermite correction, deterministic shooting, and MCMC-based path sampling. Each of these schemes is implemented using the Milstein discretization with local correction terms that preserve higher-order stochastic moments. The Hermite-corrected bridge, in particular, incorporates cubic and quartic terms of the transition density expansion, providing an enhanced representation of curvature and skewness in the conditional dynamics. A detailed numerical analysis is presented for both one-dimensional and two-dimensional diffusions with nonlinear drift functions and spatially varying diffusion matrices. The examples include models with polynomial, hyperbolic, arctangent, and tangent-type drifts that exhibit strong nonlinearity and coupling. Statistical diagnostics such as effective sample size (ESS), variance of log-weights, maximum normalized share, and perplexity are used to assess the efficiency of the importance reweighting. Residual-based QQ plots and histograms confirm that the Gaussian-Hermite bridge maintains near-Gaussian residual distributions and stable log-weight statistics across a broad range of time horizons. The obtained results demonstrate that the Gaussian-Hermite decomposition significantly improves the stability, accuracy, and efficiency of stochastic bridge simulations compared to Gaussian or shooting-based methods. The approach provides a theoretically consistent and computationally tractable framework for conditioned diffusion processes with low-regularity coefficients. Beyond SBVPs, the method can be extended to parameter estimation, inverse problems, and stochastic control formulations, where explicit transition densities or their functional decompositions are available.
This paper introduces a novel estimator for the spectral density function, combining the quantile periodogram and the multitaper periodogram. We demonstrate that the quantile multitaper periodogram inherits the robustness properties of the quantile periodogram while benefiting from the bias and variance reduction achieved through multitapering. The proposed spectral estimator is well-suited for time series analysis under general conditions of non-linearity and non-normality.
This article proposes a method for constructing computer experiment designs, specifically for Strauss processes marked by three types of marks. This method is based on the theory of marked point processes. The designs obtained using the Monte Carlo Markov Chain (MCMC) method and the Metropolis-Hastings algorithm are easily adaptable and can meet various objectives. A detailed study on the convergence of the Markov chain was conducted. In addition, a comparison was made between our approach and other existing computer experiment designs.
Low discrepancy sequences (sometimes called "quasi-Monte Carlo" or "sub-random" sequences) are often used to try to improve the convergence properties of multi-dimensional Monte Carlo simulation exercises. These sequences aim to sample the relevant multi-dimensional domain space of the simulation exercise more uniformly than the basic Monte Carlo approach of choosing sequence elements purely at random. However, the convergence improvements provided by traditional low discrepancy sequence methodologies such as Halton or Sobol' sequences can seem puzzlingly intermittent. We explore why this is so. We also propose an approach that addresses many of the relevant issues. It involves defining in advance the targeted length of the sequence and then optimising how sequence elements are selected bearing in mind this target length. A particularly appealing example involves a sequence length that is an integer power of two and choosing sequence elements using a method similar to that used to generate Sobol' sequences but with scrambling and length-specific overlays.
This work addresses a well-known problem in Monte Carlo methods: The estimation of small probabilities for diffusion particles to reach small boundary regions located far from point sources. The problem is solved using a new stochastic method based on a combination of the global Random Walk on Spheres (RWS) algorithm and an iterative refinement applied directly to the underlying differential problem. The problems are studied in stationary formulation with mixed Dirichlet and Neumann boundary conditions. A series of numerical experiments demonstrates the superiority of the proposed algorithms over other approaches, in particular over methods based on fundamental solutions.
This paper introduces the versatile double-sided rejection method for simulating one-dimensional random variables. Effective for variables with continuous monotone densities over bounded intervals, the method uses piecewise constant majorants and minorants. The new technique for probability equalization is developed, involving constraction of special non-uniform meshes. The custom-built computer system PrEMA (Probability Equalization Modelling Algorithms) - available at https://prema.andronix1.ru - is presented. PrEMA includes the "distributed-random" library with C and Rust code implementations of the method. It also features a dialog system for comparing the costs of the double-sided algorithm to the inverse distribution function method and the modified ziggurat algorithm. The impact of different standard pseudo-random number generators on the system's performance is examined.
In this paper, we show that we can generate nonrecursive n-bit pseudorandom numbers by two times (n-bit) x (n-bit) multiplication and succeeding extraction of an n-bit integer from the result of multiplication. The algorithm, which we call MmS, is described using the functions defined by T-2k(X, Y) = 2(k) XY - left perpendicular2(k) XYright perpendicular + 1, X, Y is an element of[1, 2), and T-2k(X) equivalent to T-2k(X, X). We observe and consider mathematically the condition that T-2k(X, Y) and succeeding repetition of T-2k(X) generates random numbers, and see why our algorithm MmS is independent of the value of n. We also show the way of reducing the times of multiplication from two times to one time.
We show that Lasso and Bayesian Lasso are very close when the sparsity is large and the noise is small. We propose to solve Bayesian Lasso using multivalued stochastic differential equation. We derive four discretization algorithms, and present highly efficient multilevel Monte Carlo (MLMC) simulations. Additionally, we perform a numerical comparison of the Monte Carlo (MC), MLMC and proximal Markov chain Monte Carlo algorithm (PMALA).
Deeply virtual exclusive scattering processes (DVES) serve as precise probes of nucleon quark and gluon distributions in coordinate space. These distributions are derived from generalized parton distributions (GPDs) via Fourier transform relative to proton momentum transfer. QCD factorization theorems enable DVES to be parameterized by Compton form factors (CFFs), which are convolutions of GPDs with perturbatively calculable kernels. Accurate extraction of CFFs from DVCS, benefiting from interference with the Bethe-Heitler (BH) process and a simpler final state structure, is essential for inferring GPDs. This paper focuses on extracting CFFs from DVCS data using a variational autoencoder inverse mapper (VAIM) and its constrained variant (C-VAIM). VAIM is shown to be consistent with Markov Chain Monte Carlo (MCMC) methods in extracting multiple CFF solutions for given kinematics, while C-VAIM effectively captures correlations among CFFs across different kinematic values, providing more constrained solutions. This study represents a crucial first step towards a comprehensive analysis pipeline towards the extraction of GPDs.
In this paper, we study the class of first order Periodic Generalized Autoregressive Conditional heteroscedasticity processes (PGARCH(1, 1) for short) in which the parameters in the volatility process are allowed to switch between different regimes. First, we establish necessary and sufficient conditions for a PGARCH(1, 1) process to have a unique stationary solution (in periodic sense) and for the existence of moments of any order. Next, we are able to estimate the unknown parameters involved in model via the so-called generalized method of moments (GMM). We construct an estimator and establish its asymptotic properties. Specifically, we demonstrate its consistency and asymptotic normality. The GMM estimator in some cases can be more efficient than the least squares estimator (LSE) and the quasi maximum likelihood estimator (QMLE). Some simulation studies are also performed to highlight the impact of our theoretical results.
This study develops explicit algebraic expressions for the single and product moments of order statistics derived from the generalized Bilal (GB) distribution. These expressions facilitate the computation of means, variances and covariances of order statistics for sample sizes up to n = 10 {n=10} with specified parameter values. The derived moments serve as the foundation for constructing the best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) for the location and scale parameters applicable to both complete and type-II right censored samples. Additionally, the study explores the prediction of unobserved order statistics in type-II right censored samples. The theoretical results are validated through a simulation study, while a real data example highlights their practical utility. These findings establish a robust framework for statistical inference based on order statistics from the GB distribution.
The permuted congruential generators is a set of pseudorandom number generators released by Melissa E. O'Neill in 2014. The original technical report outlined several lightweight scrambling techniques designed for the linear congruential generator. Each scrambling technique offered some improvement to the quality of the linear congruential generator. However, the real strength of the scrambling techniques was that they could be combined into multiple overall stronger scramblers. The technical report concludes with the creation of the PCG library, a popular pseudorandom number generation library that implements several generators described in the technical report. Starting from the observation that the paper's work was narrowly focused on implementing their scrambling techniques for specific linear congruential generators, we explore the permuted congruential generator scrambling techniques and their potential for being applied to other pseudorandom number generators by generalizing the scrambling techniques to work across different pseudorandom number generators.
This work suggests different Monte Carlo algorithms for solving large systems of linear algebraic equations arising from the numerical solution of the Dirichlet problem for the Helmholtz equation. Approach based on boundary integral representations, vector randomization algorithm, method of fundamental solutions, stochastic projection algorithm, and randomized singular value decomposition are applied. It is shown that the use of stochastic iterative refinement and preconditioning can significantly improve the accuracy and stability of the computations. Simulation results are presented, demonstrating the effectiveness of the proposed methods.
We revisit the method of antithetic variates, perhaps the simplest yet profound variance reduction technique among many others, with the aim of comprehensively improving the method in its general formulation in the seminal works A new Monte Carlo technique: Antithetic variates [Hammersley and Morton, Math. Proc. Cambridge Philos. Soc. 52 (1956), 449–475] and Antithetic variates revisited [Fishman and Huang, Commun. ACM 26 (1983), 964–971]. The achieved advancement under this general formulation contrasts with the conventional approach based on equally weighted two variates with negative correlation, commonly referred to as the method of antithetic variates in various contexts. In pursuit of the aim, we investigate its second-order structure in depth and introduce an adaptive algorithm designed to optimize the weights among multiple variates throughout the primary Monte Carlo estimation process. In order to effectively demonstrate the theoretical advancements, we present numerical results throughout the presentation, vividly showcasing the potential effectiveness of the proposed approach and adaptive algorithm in the face of varying weights.
In this note we will discuss and review some recent results concerning well-known financial mathematical problems such as portfolio construction, dynamic trading, option pricing, etc. We will use some Python codes concerning the above and we will compare the results with the existing methods and techniques. We propose also a new type of multi-asset options; the options on correlation. Using this kind of options one can refine more effectively the profit function of his/her portfolio when this contain two or more assets. It is mathematically certain that, in practice, someone will eventually apply the techniques described in this paper. This is because, regarding the portfolio construction problem, we allow the investor to employ any forecasting technique – e.g., statistical methods, machine learning, behavioral finance, intuition, etc. Subsequently, the investor can enhance both the return and safety of their portfolio by incorporating call and put options. In contrast, for the derivative pricing problem, it is evident that there is no room for forecasts (see volatility for example), as pricing involves two counterparties – the seller and the buyer – making it, metaphorically, a dance for two. For this reason, the pricing methodology proposed in this paper is model-free, ensuring that the resulting prices are fully consistent with the market values of available contracts. Moreover, the investor decides at which price to buy or sell a contract based on practical, statically implementable hedging strategies that we propose in this work. That is, any pricing method should justify why an investor ought to buy or sell a derivative at the proposed price. In other words, the method must provide a clear, economically sound rationale-typically grounded in no-arbitrage principles, replication arguments, or explicit hedging strategies – that links the quoted price to actionable, implementable decisions for market participants. What remains to be explored are advanced forecasting techniques that account for events affecting the stocks of interest to the investor, as well as the documentation of hedging strategies for path-dependent options.
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) x (n-bit) multiplication and taking out an n-bit integer from the result of multiplication. The algorithm can be described by beta-transformation T-beta(X) = beta X - left perpendicular beta Xright perpendicular + 1, X is an element of [1, 2), beta > 1. We consider the condition that repetition of beta-transformation generates random numbers, and see why our simple algorithm works well for various values of n
