Euler-Maruyama Research Articles

Bass Construction with Multi-Marginals: Lightspeed Computation in a New Local Volatility Model

The local volatility model is widely used as this is the unique one-factor Markov model perfectly calibrated to a continuum of vanilla options in strike and expiry. It requires unfortunately an arbitrage-free interpolation of implied volatility in expiry and a time-consuming Euler discretization scheme for its simulation. In this paper, we construct a new local volatility model, based on the extension of the Bass construction, which is (1) perfectly calibrated to vanilla options on market expiries and (2) is also a one-factor diffusion which can be discretized exactly as it requires only the simulation of a standard Brownian motion, providing very fast calculations.

Compensated two-step Maruyama methods for stochastic differential equations with Poisson jumps

In this article we present the compensated two-step Maruyama methods for the stochastic differential equations with Poisson jumps. Mean-square (M-S) order convergence is established and M-S stability analysis are displayed. In particular, compensated two-step Maruyama methods of Adams type are considered, their linear M-S stability regions compared with compensated Euler–Maruyama method (EM) are plotted. Numerical results show the M-S convergence and stability are given.

A cubic nonlinear population growth model for single species: theory, an explicit\u2013implicit solution algorithm and applications

In this paper, we extend existing population growth models and propose a model based on a nonlinear cubic differential equation that reveals itself as a special subclass of Abel differential equations of first kind. We first summarize properties of the time-continuous problem formulation. We state the boundedness, global existence, and uniqueness of solutions for all times. Proofs of these properties are thoroughly given in the Appendix to this paper. Subsequently, we develop an explicit–implicit time-discrete numerical solution algorithm for our time-continuous population growth model and show that many properties of the time-continuous case transfer to our numerical explicit–implicit time-discrete solution scheme. We provide numerical examples to illustrate different behaviors of our proposed model. Furthermore, we compare our explicit–implicit discretization scheme to the classical Eulerian discretization. The latter violates the nonnegativity constraints on population sizes, whereas we prove and illustrate that our explicit–implicit discretization algorithm preserves this constraint. Finally, we describe a parameter estimation approach to apply our algorithm to two different real-world data sets.

Bifurcation and chaos analysis for a discrete ecological developmental systems.

This work concentrates on the dynamic analysis including bifurcation and chaos of a discrete ecological developmental systems. Specifically, it is a prey–predator–scavenger (PPS) system, which is derived by Euler discretization method. By choosing the step size h as a bifurcation parameter, we determine the set consists of all system’s parameters, in which the system can undergo flip bifurcation (FB) and Neimark–Sacker bifurcation (NSB). The theoretical results are verified by some numerical simulations. It is shown that the discrete systems exhibit more interesting behaviors, including the chaotic sets, quasi-periodic orbits, and the cascade of period-doubling bifurcation in orbits of periods 2, 4, 8, 16. Finally, corresponding to the two bifurcation behaviors discussed, the maximum Lyapunov exponent is numerically calculated, which further verifies the rich dynamic characteristics of the discrete system.

Exponential stability of non-linear neutral stochastic delay differential system with generalized delay-dependent impulsive points

This work aims to analyze the exponential stability of a non-linear impulsive neutral stochastic delay differential system. In this study, impulse perturbation is considered a delay-dependent state variable. The solution of the delay-dependent impulsive neutral stochastic delay differential system is associated with the solution of the system without impulses. First, we developed a relation connecting the solution of the neutral stochastic delay differential system without impulses and the solution of the corresponding system with impulses. Then, the conditions of the exponential stability of the proposed impulsive system are derived by determining the stability analysis of the respective system without impulse. The numerical approach for the neutral stochastic delay system without impulses is generated using the Euler-Maruyama method and adopted for the corresponding impulsive system. Finally, the achieved theoretical results are illustrated for applying the Malthusian single species neutral stochastic delay population model with immigration impulses.

Strong convergence rate of truncated Euler-Maruyama method for stochastic differential delay equations with Poisson jumps.

We study a class of super-linear stochastic differential delay equations with Poisson jumps (SDDEwPJs). The convergence and rate of the convergence of the truncated Euler-Maruyama numerical solutions to SDDEwPJs are investigated under the generalized Khasminskii-type condition.

Implicit Discrete-Time Adaptive First-Order Sliding Mode Control With Predefined Convergence Time

This brief proposes an adaptive first-order sliding mode control (SMC) with a predefined convergence time in a digital framework. It is based on an implicit Euler discretization of the set-valued signum function in SMC and a time-dependent adaptive gain scheme. Two typical cases are discussed and their stability analysis is provided, simple affine systems with first-order SMC and LTI systems with equivalent control-based sliding mode control (ECB-SMC), both in the presence of unknown perturbations. Numerical simulations show that, compared with the conventional SMC, the proposed implicit discrete-time SMC guarantees the convergence of the states within any predefined arbitrary time.

Dynamic Euler Discretization of Exploding Processes

We propose a new numerical method to accurately simulate the trajectory of stochastic differential equations with exploding coefficients. The approach relies on the Euler method, where we dynamically choose the step size to obtain the desired root-mean-square error. To determine the step size, we rely on the toy model of the explosion model in which an analytical solution is available. We demonstrated the superiority of the dynamic Euler approach compared to the fixed step size method (standard Euler method) based on simulation experiments.

Invariance under discretization for positive systems

Positive dynamical or control systems have all their variables nonnegative. Euler discretization transforms a continuous-time system into a system on a discrete time scale. Some structural properties of the system may be preserved by discretization, while other may be lost. Four fundamental properties of positive systems are studied in the context of discretization: positivity, positive stability, positive reachability and positive observability. Both linear and nonlinear systems are investigated.

Positivity preserving truncated Euler–Maruyama Method for stochastic Lotka–Volterra competition model

The well-known stochastic Lotka–Volterra model for interacting multi-species in ecology has some typical features: highly nonlinear, positive solution and multi-dimensional. The known numerical methods including the tamed/truncated Euler–Maruyama (EM) applied to it do not preserve its positivity. The aim of this paper is to modify the truncated EM to establish a new positive preserving truncated EM (PPTEM). To simplify the proof as well as to make our theory more understandable, we will first develop a nonnegative preserving truncated EM (NPTEM) and then establish the PPTEM. Of course, we should point out that the NPTEM has its own right as many SDE models in applications have their nonnegative solutions.

Convergence and positivity of Finite Element methods for a haptotaxis model of tumoral invasion

In this paper, we consider a mathematical model for the invasion of host tissue by tumor cells in a d-dimensional bounded domain, d≤3. This model consists of a system of differential equations describing the evolution of cancer cell density, the extracellular matrix protein density and the matrix degrading enzyme concentration. We develop two fully discrete schemes for approximating the solutions based on a semi-implicit Euler discretization in time and Finite Element (FE) discretization on space (restricted to triangularization made up of right-angled simplices) of two equivalent systems. For the first numerical scheme, we use a splitting technique to deal with the haptotaxis term, leading to introduce an equivalent system with a new variable given by the gradient of extracellular matrix. This scheme is well-posed and preserves the non-negativity of extracellular matrix and the degrading enzyme. We analyze error estimates and convergence towards regular solutions. The second numerical scheme is based on an equivalent formulation in which the cancer cell density equation is expressed in divergence form through a suitable change of variables. This second numerical scheme preserves the non-negativity of all the discrete variables. Finally, we present some numerical simulations in agreement with the theoretical analysis.

New finite volume element methods in the ALE framework for time-dependent convection–diffusion problems in moving domains

This paper develops new finite volume element methods (FVEMs) in the Arbitrary Lagrangian–Eulerian (ALE) framework for the time-dependent convection–diffusion problems on moving domains. In particular, we present two fully discrete schemes, one is based on the implicit Euler (IE) discretization and the other is based on the combination of the IE and geometric conservation laws (GCL). Stability and error estimation are analyzed for these two schemes. The second scheme satisfying the GCL demonstrates better on stability for long time simulations. Finally, numerical experiments are carried out to illustrate the theoretical results.

A stochastic differential equation approach to the analysis of the 2017 and 2019 UK general election polls

Human dynamics and sociophysics build on statistical models that can shed light on and add to our understanding of social phenomena. We propose a generative model based on a stochastic differential equation that enables us to model the opinion polls leading up to the 2017 and 2019 UK general elections and to make predictions relating to the actual results of the elections. After a brief analysis of the time series of the poll results, we provide empirical evidence that the gamma distribution, which is often used in financial modelling, fits the marginal distribution of this time series. We demonstrate that the proposed poll-based forecasting model may improve upon predictions based solely on polls. The method uses the Euler–Maruyama method to simulate the time series, measuring the prediction error with the mean absolute error and the root mean square error, and as such could be used as part of a toolkit for forecasting elections.

Biorthogonal Wavelet on a Logarithm Curve ℂ

According to the length-preserving projection and Euler discretization method, biorthogonal wavelet function on a smooth curve C is constructed in this paper, such as a logarithm curve. The properties of biorthogonal wavelet filters on a smooth curve C are discussed, such as induced refinable equation and symmetry. Moreover, an example is given for discussing the biorthogonal scaling function and its dual on a logarithm curve C . Finally, a numerical application is given for dealing with financial data.

Classification and image processing with a semi‐discrete scheme for fidelity forced Allen–Cahn on graphs

AbstractThis paper introduces a semi‐discrete implicit Euler (SDIE) scheme for the Allen‐Cahn equation (ACE) with fidelity forcing on graphs. The continuous‐in‐time version of this differential equation was pioneered by Bertozzi and Flenner in 2012 as a method for graph classification problems, such as semi‐supervised learning and image segmentation. In 2013, Merkurjev et. al. used a Merriman‐Bence‐Osher (MBO) scheme with fidelity forcing instead, as heuristically it was expected to give similar results to the ACE. The current paper rigorously establishes the graph MBO scheme with fidelity forcing as a special case of an SDIE scheme for the graph ACE with fidelity forcing. This connection requires the use of the double‐obstacle potential in the ACE, as was already demonstrated by Budd and Van Gennip in 2020 in the context of ACE without a fidelity forcing term. We also prove that solutions of the SDIE scheme converge to solutions of the graph ACE with fidelity forcing as the discrete time step converges to zero. In the second part of the paper we develop the SDIE scheme as a classification algorithm. We also introduce some innovations into the algorithms for the SDIE and MBO schemes. For large graphs, we use a QR decomposition method to compute an eigendecomposition from a Nyström extension, which outperforms the method used by, for example, Bertozzi and Flenner in 2012, in accuracy, stability, and speed. Moreover, we replace the Euler discretization for the scheme's diffusion step by a computation based on the Strang formula for matrix exponentials. We apply this algorithm to a number of image segmentation problems, and compare the performance with that of the graph MBO scheme with fidelity forcing. We find that while the general SDIE scheme does not perform better than the MBO special case at this task, our other innovations lead to a significantly better segmentation than that from previous literature. We also empirically quantify the uncertainty that this segmentation inherits from the randomness in the Nyström extension.

The effect of time discretization on the propeller hydrodynamic performance simulation in self-propulsion and open water conditions

Numerical simulation technology is widely used in the field of naval architect and marine engineering. The flow induced by the propeller is unsteady. In order to demonstrate the effect of time discretization format and time step on the simulation of the flow, this paper takes an oil tanker and its propeller as the research objects, and simulations have been conducted by using OpenFoam. By comparing the thrust, torque and the vortex after the propeller, the effects of the time discretization format and time step on the numerical simulations are analyzed. This paper believes that changing the discretization format of the time term while keep the time step unchanged has little effect on the calculation results in the condition of open water test simulation. It is beneficial to simulate of the vortex after the propeller hub by using a small time step. The first-order backward Euler discretization format in the condition of self propulsion simulation is recommended. If the error of thrust compared with experiment data is larger than expected, the second-order discretization format should be considered. In order to improve the accuracy of numerical simulation, a smaller time step could be considered.

Models for plasma kinetics during simultaneous therapeutic plasma exchange and extracorporeal membrane oxygenation.

This paper focuses on the derivation and simulation of mathematical models describing new plasma fraction in blood for patients undergoing simultaneous extracorporeal membrane oxygenation and therapeutic plasma exchange. Models for plasma exchange with either veno-arterial or veno-venous extracorporeal membrane oxygenation are considered. Two classes of models are derived for each case, one in the form of an algebraic delay equation and another in the form of a system of delay differential equations. In special cases, our models reduce to single compartment ones for plasma exchange that have been validated with experimental data (Randerson et al., 1982, Artif. Organs, 6, 43-49). We also show that the algebraic differential equations are forward Euler discretizations of the delay differential equations, with timesteps equal to transit times through model compartments. Numerical simulations are performed to compare different model types, to investigate the impact of plasma device port switching on the efficiency of the exchange process, and to study the sensitivity of the models to their parameters.

Stability equivalence among stochastic differential equations and stochastic differential equations with piecewise continuous arguments and corresponding Euler-Maruyama methods

In this paper, we consider the equivalence of the pth moment exponential stability for stochastic differential equations (SDEs), stochastic differential equations with piecewise continuous arguments (SDEPCAs) and the corresponding Euler-Maruyama methods EMSDEs and EMSDEPCAs. We show that if one of the SDEPCAs, SDEs, EMSDEs and EMSDEPCAs is pth moment exponentially stable, then any of them is pth moment exponentially stable for a sufficiently small step size h and τ under the global Lipschitz assumption on the drift and diffusion coefficients.

Optimal control of COVID-19

Coronavirus disease of 2019 or COVID-19 (acronym for coronavirus disease 2019) is an emerging infectious disease caused by a strain of coronavirus called SARS-CoV-22, contagious with human-to-human transmission via respiratory droplets or by touching contaminated surfaces then touching them face. Faced with what the world lives, to define this problem, we have modeled it as an optimal control problem based on the models of William Ogilvy Kermack et Anderson Gray McKendrick, called SEIR model, modified by adding compartments suitable for our study. Our objective in this work is to maximize the number of recovered people while minimizing the number of infected. We solved the problem theoretically using the Pontryagin maximum principle, numerically we used and compared results of two methods namely the indirect method (shooting method) and the Euler discretization method, implemented in MATLAB.

Sampled-Data Stabilization of a Class of Stochastic Nonlinear Markov Switching System with Indistinguishable Modes Based on the Approximate Discrete-Time Models

This paper investigates the stabilization issue for a class of sampled-data nonlinear Markov switching system with indistinguishable modes. In order to handle indistinguishable modes, the authors reconstruct the original mode space by mode clustering method, forming a new merged Markov switching system. By specifying the difference between the Euler-Maruyama (EM) approximate discrete-time model of the merged system and the exact discrete-time model of the original Markov switching system, the authors prove that the sampled-data controller, designed for the merged system based on its EM approximation, can exponentially stabilize the original system in mean square sense. Finally, a numerical example is given to illustrate the effectiveness of the method.