Stochastic Runge-Kutta Methods Research Articles

A study on generalized balanced split drift stochastic Runge- Kutta methods for stochastic differential equations

To reduce computational complexity, the balanced numerical approximations of the general split drift stochastic Runge-Kutta methods are analyzed. The primary reasons for considering the numerical approximations of these balanced split stochastic Runge-Kutta methods are their improved stability characteristics and lower mean square error compared to other methods. By balancing the drift and diffusion components, the splitting techniques outperform the mean square error over longer time increments. For Ito multi-dimensional stochastic differential equations, we propose a novel family of balanced universal split stochastic Runge-Kutta procedures. The Kronecker product concept is utilized to derive the mean-square stability conditions. We conduct numerical tests to evaluate these methods against an existing weak order 2 split drift method. Ultimately, a specific numerical example validates the theoretical outcomes of the balanced general split stochastic Runge-Kutta procedures.

Stochastic Runge-Kutta Methods for Preserving Maximum Bound Principle of Semilinear Parabolic Equations. Part I: Gaussian Quadrature Rule

Stochastic Runge-Kutta Methods for Preserving Maximum Bound Principle of Semilinear Parabolic Equations. Part I: Gaussian Quadrature Rule

Exotic B-Series and S-Series: Algebraic Structures and Order Conditions for Invariant Measure Sampling

AbstractB-Series and generalizations are a powerful tool for the analysis of numerical integrators. An extension named exotic aromatic B-Series was introduced to study the order conditions for sampling the invariant measure of ergodic SDEs. Introducing a new symmetry normalization coefficient, we analyze the algebraic structures related to exotic B-Series and S-Series. Precisely, we prove the relationship between the Grossman–Larson algebras over exotic and grafted forests and the corresponding duals to the Connes–Kreimer coalgebras and use it to study the natural composition laws on exotic S-Series. Applying this algebraic framework to the derivation of order conditions for a class of stochastic Runge–Kutta methods, we present a multiplicative property that ensures some order conditions to be satisfied automatically.

Role of asymmetry and external noise in the development and synchronization of oscillations in the analog Hopfield neural networks with time delay.

The analog Hopfield neural network with time delay and random connections has been studied for its similarities in activity to human electroencephalogram and its usefulness in other areas of the applied sciences such as speech recognition, image analysis, and electrocardiogram modeling. Our goal here is to understand the mechanisms that affect the rhythmic activity in the neural network and how the addition of a Gaussian noise contributes to the network behavior. The neural network studied is composed of ten identical neurons. We investigated the excitatory and inhibitory networks with symmetric (square matrix) and asymmetric (triangular matrix) connections. The differential equations that model the network are solved numerically using the stochastic second-order Runge-Kutta method. Without noise, the neural networks with symmetric and asymmetric matrices possessed different synchronization properties: fully connected networks were synchronized both in time and in amplitude, while asymmetric networks were synchronized in time only. Saturation outputs of the excitatory neural networks do not depend on the time delay, whereas saturation oscillation amplitudes of inhibitory networks increase with the time delay until the steady state. The addition of the Gaussian noise is shown to significantly amplify small-amplitude oscillations, dramatically accelerates the rate of amplitude growth to saturation, and changes synchronization properties of the neural network outputs.

DEVELOPMENT OF ALGORITHMS AND SOFTWARE FOR MODELING CONTROLLED DYMAMIC SYSTEMS USING SYMBOLIC COMPUTATIONS AND STOCHASTIC METHODS

The development of software for synthesizing and analyzing models of controlled systems taking into account their deterministic and stochastic description is an important direction of research. Results of the development of software for modeling dynamic systems the behavior of which can be described by onestep processes are presented. Models of population dynamics are considered as an example. The software uses a deterministic description of the model at its input to obtain a corresponding stochastic model in symbolic form and also analyze the model in detail (calculate trajectories in the deterministic and stochastic cases, find control functions, and visualize the results). An important aspect of the development is the use of computer algebra for analyzing the model and synthesizing controls. Methods and algorithms based on deterministic and stochastic Runge–Kutta methods, stability and control theory, methods for designing self-consistent stochastic models, numerical optimization algorithms, and artificial intelligence are implemented. The software was developed using high-level programming languages Python and Julia. As the basic tools, high-performance libraries for vector–matrix computations, symbolic computation libraries, libraries for the numerical solution of ordinary differential equations, and libraries of global optimization algorithms are used.

Fractional discrete Temimi–Ansari method with singular and nonsingular operators: applications to electrical circuits

The goal of this article is to present a recently developed numerical approach for solving fractional stochastic differential equations with a singular Caputo kernel and a nonsingular Caputo–Fabrizio and Atangana–Baleanu (ABC) kernel. The proposed method is based on the discrete Temimi–Ansari method, which is combined with three different numerical schemes that are appropriate for the new fractional derivative operators. The proposed technique is used to investigate the effects of Gaussian white-noise and Gaussian colored-noise perturbations on the potential source and resistance in fractional stochastic electrical circuits. The proposed method’s robustness and efficiency were demonstrated by comparing its results to those of the stochastic Runge–Kutta method (SRK). The valuable point in this article is that the resulting numerical scheme is able to combine two powerful methods that can be extended into more complex stochastic models. The comparison of different fractional derivatives using Mathematica 12 software has been obtained and the simulation results demonstrate the merit of the contributed method.

Weak convergence of balanced stochastic Runge–Kutta methods for stochastic differential equations

In this paper, weak convergence of balanced stochastic one-step methods and especially balanced stochastic Runge–Kutta (SRK) methods for Itô multidimensional stochastic differential equations is analyzed. Generalizing a corresponding result obtained by H. Schurz for the standard Euler method, it is shown that under certain conditions, balanced one-step methods preserve the weak convergence properties of their underlying methods. As an application, this allows to prove the weak convergence order of the balanced SRK methods presented in earlier work by A. Rathinasamy, P. Nair and D. Ahmadian.

Conservative Continuous-Stage Stochastic Runge–Kutta Methods for Stochastic Differential Equations

In this paper, we develop a new class of conservative continuous-stage stochastic Runge–Kutta methods for solving stochastic differential equations with a conserved quantity. The order conditions of the continuous-stage stochastic Runge–Kutta methods are given based on the theory of stochastic B-series and multicolored rooted tree. Sufficient conditions for the continuous-stage stochastic Runge–Kutta methods preserving the conserved quantity of stochastic differential equations are derived in terms of the coefficients. Conservative continuous-stage stochastic Runge–Kutta methods of mean square convergence order 1 for general stochastic differential equations, as well as conservative continuous-stage stochastic Runge–Kutta methods of high order for single integrand stochastic differential equations, are constructed. Numerical experiments are performed to verify the conservative property and the accuracy of the proposed methods in the longtime simulation.

Adaptive Step Size Stochastic Runge-Kutta Method of Order 1.5(1.0) for Stochastic Differential Equations (SDEs)

The stiff stochastic differential equations (SDEs) involve the solution with sharp turning points that permit us to use a very small step size to comprehend its behavior. Since the step size must be set up to be as small as possible, the implementation of the fixed step size method will result in high computational cost. Therefore, the application of variable step size method is needed where in the implementation of variable step size methods, the step size used can be considered more flexible. This paper devotes to the development of an embedded stochastic Runge-Kutta (SRK) pair method for SDEs. The proposed method is an adaptive step size SRK method. The method is constructed by embedding a SRK method of 1.0 order into a SRK method of 1.5 order of convergence. The technique of embedding is applicable for adaptive step size implementation, henceforth an estimate error at each step can be obtained. Numerical experiments are performed to demonstrate the efficiency of the method. The results show that the solution for adaptive step size SRK method of order 1.5(1.0) gives the smallest global error compared to the global error for fix step size SRK4, Euler and Milstein methods. Hence, this method is reliable in approximating the solution of SDEs.

Stochastic Bayesian Runge-Kutta method for dengue dynamic mapping

Dengue Hemorrhagic Fever (DHF) is still a threat to humanity that cause death and disability due to changes in environmental and socioeconomic conditions, especially in tropical areas. A critical assessment of the models and methods is necessary. The vital role of stochastic processes of infectious disease research in Bayesian statistical models helps provide an explicit framework for understanding disease transmission dynamics between hosts (humans) and vectors (mosquitoes). This research presents a Bayesian stochastic process for the cross-infection SIR-SI model as a differential equation (susceptibility–infection–recovery for the human population; susceptible–infectious for the vector population) for the dynamic transmission of DHF. Given the difficulties of solving the differential equations precisely, we propose a computational model to approach the solution based on the Runge-Kutta family approach, namely the Euler and the four-order Runge-Kutta methods. The methods used for the discretization process of the SIR-SI model for computational process needs. For comparison purposes, we use a monthly DHF dataset of 10 Kendari, Indonesia, districts from 2019 to 2021. Parameter estimation of the Bayesian SIR-SI model based on the Euler and four-order Runge-Kutta method was updated using Markov Chain Monte Carlo (MCMC) Bayesian. The Euler and four-order Runge-Kutta methods have converged at 10,000 iterations with burn-in 80,000. The numerical simulation results show that the four-order Runge-Kutta approach has the slightest deviance, 106.5. Therefore, this approach is the best one. The relative risk analysis shows that the dynamics of DHF cases fluctuate from January to July every year. However, from January to May, there was a high consistency of DHF cases. Two districts with high case consistency were found, namely Kadia and Wua-Wua. Furthermore, because the spread of DHF cases has a spatial effect, the Kadia and Wua-Wua districts need serious attention to suppress the rate of reach of DHF in Kendari City. Intensive observation and intervention in the Kadia and Wua-Wua districts should be carried out in early January to stop the breeding of mosquito larvae. In other neighbourhoods such as Puwatu, Kambu and Kendari Barat, the DHF cases occur temporally as the impacts of DHF cases in the Kadia and Wua-Wua districts. It may facilitate the statisticians to develop further models to adopt a better understanding to control the DHF dynamically. In brief,

Explicit pseudo-symplectic Runge-Kutta methods for stochastic Hamiltonian systems

We give conditions for stochastic Runge-Kutta methods to near preserve quadratic invariants, and we discuss the associated simplified order conditions. For stochastic Hamiltonian systems we propose a systematic approach to construct explicit stochastic Runge-Kutta pseudo-symplectic schemes. Our approach is based on colored trees and B-series. We construct some pseudo-symplectic stochastic Runge-Kutta methods with strong convergence order, and we illustrate numerically the long term performance of the proposed schemes.

Stochastic Capital–Labor Lévy Jump Model with the Precariat Labor Force

In this work, we study a capital–labor model by considering the interaction between the new proposed and the confirmed free jobs, the precariat labor force, and the mature labor force by introducing Brownian motion and Lévy noise. Moreover, we illustrate the well-posedness of the solution. In addition, we establish the conditions of the extinction of both the free jobs and labor force; subsequently, we prove the persistence of only the free jobs, and we also show the conditions of the persistence of both the free jobs and labor force. Finally, we validate our theoretical finding by numerical simulation by building a new stochastic Runge–Kutta method.

Modelo con elasticidad constante de la varianza: Una aplicación en la Bolsa de Valores de Lima

The objective of this article is to present the constant elasticity of variance model, its main probabilistic aspects, and we implement the stochastic Runge Kutta method to simulate the sample trajectories based on the data of the CREDITC1 asset of the Bank of Credit from Perú taken from the Lima stock exchange.

The two-particle irreducible effective action for classical stochastic processes

By combining the two-particle-irreducible (2PI) effective action common in non-equilibrium quantum field theory with the classical Martin–Siggia–Rose formalism, self-consistent equations of motion for the first and second cumulants of non-linear classical stochastic processes are constructed. Such dynamical equations for correlation and response functions are important in describing non-equilibrium systems, where equilibrium fluctuation–dissipation relations are unavailable. The method allows to evolve stochastic systems from arbitrary Gaussian initial conditions. In the non-linear case, it is found that the resulting integro-differential equations can be solved with considerably reduced computational effort compared to state-of-the-art stochastic Runge–Kutta methods. The details of the method are illustrated by several physical examples.

Two semi-analytical approaches to approximate the solution of stochastic ordinary differential equations with two enormous engineering applications

This study proposes a new large-scale application for efficient semi-analytical techniques, Variation of Parameters Method with an auxiliary parameter and Temimi and Ansari method. In this regard, Variation of Parameters Method with auxiliary parameter and Temimi and Ansari method are advocated to sacrifice an important class of stochastic ordinary differential equations. Two enormous engineering applications with stochastic excitations are solved and compared with the results of the stochastic Runge Kutta method (SRK) to crystallize the accuracy of the proposed methods. Also, a comparison between the two proposed methods is presented. The numerical results demonstrate that Temimi and Ansari method is accurate and readily implemented. It is also worth noting that the proposed two methods have the advantage of easily implantation without linearization or perturbation.

Higher strong order methods for linear Itô SDEs on matrix Lie groups

In this paper we present a general procedure for designing higher strong order methods for linear Itô stochastic differential equations on matrix Lie groups and illustrate this strategy with two novel schemes that have a strong convergence order of 1.5. Based on the Runge–Kutta–Munthe–Kaas (RKMK) method for ordinary differential equations on Lie groups, we present a stochastic version of this scheme and derive a condition such that the stochastic RKMK has the same strong convergence order as the underlying stochastic Runge–Kutta method. Further, we show how our higher order schemes can be applied in a mechanical engineering as well as in a financial mathematics setting.

Extinction and stationary distribution of a stochastic COVID-19 epidemic model with time-delay

We reformulate a stochastic epidemic model consisting of four human classes. We show that there exists a unique positive solution to the proposed model. The stochastic basic reproduction number R0s is established. A stationary distribution (SD) under several conditions is obtained by incorporating stochastic Lyapunov function. The extinction for the proposed disease model is obtained by using the local martingale theorem. The first order stochastic Runge-Kutta method is taken into account to depict the numerical simulations.

Stochastic Runge–Kutta methods for multi-dimensional Itô stochastic differential algebraic equations

In this paper, we discuss the numerical solutions to index 1 stochastic differential algebraic equations. We introduce a new class of weak second-order stochastic Runge–Kutta methods for finding the numerical approximate solutions to multi-dimensional stochastic differential algebraic equations. A four-stage stiffly accurate stochastic Runge–Kutta methods for approximating analytical solutions to index 1 stochastic differential algebraic equations are derived. By colored rooted tree analysis, the order conditions for the stochastic Runge–Kutta methods of order two satisfying the weak convergence is obtained. The scalar test equations are considered to obtain the mean-square stability and the T-stability of weak second-order stochastic Runge–Kutta methods. Finally, some numerical illustrations are provided to prove the theoretical findings.

Positivity-preserving symplectic methods for the stochastic Lotka–Volterra predator-prey model

In this paper, positivity-preserving symplectic numerical approximations are investigated for the 2d-dimensional stochastic Lotka–Volterra predator-prey model driven by multiplicative noises, which plays an important role in ecosystem. The model is shown to possess both a unique positive solution and a stochastic symplectic geometric structure, and hence can be interpreted as a stochastic Hamiltonian system. To inherit the intrinsic biological characteristic of the original system, a class of stochastic Runge–Kutta methods is presented, which is proved to preserve positivity of the numerical solution and possess the discrete stochastic symplectic geometric structure as well. Uniform boundedness of both the exact solution and the numerical one are obtained, which are crucial to derive the conditions for convergence order one in the \(\mathbb {L}^1(\varOmega )\)-norm. Numerical examples illustrate the stability and structure-preserving property of the proposed methods over long time.

Stochastic modelling of the oral cancer proliferation and death in the presence of Thymoquinone as anticancer therapeutics

Oral cancer is one of the most commonly known cancer worldwide. Thymoquinone (TQ) an extract from Nigella sativa, has clinically been proven as an anticancer therapeutic agent for oral cancer due to its intrinsic pharmacological characteristics. Understanding the mechanisms of oral cancer proliferation and death in the presence of TQ is crucial so that the insight of the interaction of cancer cells and TQ can be discovered. Cancer cells in the presence of TQ is subjected to the uncontrolled factors of the environmental noise. Deterministic model is inadequate to explain this behaviour. Herein, a stochastic model is proposed to illustrate the dynamics of HSC-3 oral cancer cell lines in the presence of TQ. The deterministic model is perturbed with the noisy behaviour which then leads to the stochastic model. The model is simulated by using a four-stage stochastic Runge-Kutta (SRK4) method and the kinetic parameters are estimated by using the maximum likelihood estimation (MLE) method. The prediction quality of the model is measured by using root mean square error (RMSE). The low values of RMSE show the best-fit of the stochastic model.