Euler-Maruyama Research Articles

Pengaruh Inflasi terhadap Strategi Optimal Investasi dan Konsumsi dengan Model Stokastik

The aim of this study is to investigate an optimal investment-consumption strategy under inflation rate with interest rate is described by Cox-Ingersol-Ross (CIR) model and volatility of the stock price is defined by Heston’s volatility model. A dynamic programming principle is used to obtain a Hamilton Jacobi Bellman (HJB) equation for the value function and choose a power utility function as utility function. The explicit solution of optimal investment and consumption are acquired with using separate variable and approach variable technique. The parameter’s values are approached by Euler-Maruyama method and Ordinary Least Square (OLS) method. Assumed that the portfolio of the investor contains a risk-free asset and a risk asset. Monthly historical data of TLK stock is used as risk asset and monthly historical data of BI 7-Day (Reverse) Repo Rate (BI7DRR) is used as risk-free asset, we obtain that the proportion of investment in stock is directly proportional to return of stock and the inflation rate does not have an impact on proportion investment in the stock. Meanwhile the optimal consumption of wealth is directly proportional to investor’s wealth and inversely proportional with inflation rate, which is the investor should consume less money of his wealth when the inflation rate increases.

Backward Euler–Maruyama Method for the Random Periodic Solution of a Stochastic Differential Equation with a Monotone Drift

In this paper, we study the existence and uniqueness of the random periodic solution for a stochastic differential equation with a one-sided Lipschitz condition (also known as monotonicity condition) and the convergence of its numerical approximation via the backward Euler–Maruyama method. The existence of the random periodic solution is shown as the limit of the pull-back flows of the SDE and the discretized SDE, respectively. We establish a convergence rate of the strong error for the backward Euler–Maruyama method with order of convergence 1/2.

A fast Euler-Maruyama method for fractional stochastic differential equations

A fast Euler-Maruyama method for fractional stochastic differential equations

Central limit theorem and self-normalized Cramér-type moderate deviation for Euler-Maruyama scheme

We consider a stochastic differential equation and its Euler-Maruyama (EM) scheme, under some appropriate conditions, they both admit a unique invariant measure, denoted by π and πη respectively (η is the step size of the EM scheme). We construct an empirical measure Πη of the EM scheme as a statistic of πη, and use Stein’s method developed in Fang, Shao and Xu (Probab. Theory Related Fields 174 (2019) 945–979) to prove a central limit theorem of Πη. The proof of the self-normalized Cramér-type moderate deviation (SNCMD) is based on a standard decomposition on Markov chain, splitting η−1/2(Πη(.)−π(.)) into a martingale difference series sum Hη and a negligible remainder Rη. We handle Hη by the time-change technique for martingale, while prove that Rη is exponentially negligible by concentration inequalities, which have their independent interest. Moreover, we show that SNCMD holds for x=o(η−1/6), which has the same order as that of the classical result in Shao (J. Theoret. Probab. 12 (1999) 385–398), Jing, Shao and Wang (Ann. Probab. 31 (2003) 2167–2215).

High-order space–time finite element methods for the Poisson–Nernst–Planck equations: Positivity and unconditional energy stability

We present a novel class of high-order space–time finite element schemes for the Poisson–Nernst–Planck (PNP) equations. We prove that our schemes are mass conservative, positivity preserving, and unconditionally energy stable for any order of approximation. To the best of our knowledge, this is the first class of (arbitrarily) high-order accurate schemes for the PNP equations that simultaneously achieve all these three properties.This is accomplished via (1) using finite elements to directly approximate the so-called entropy variableui≔U′(ci)=log(ci) instead of the density variable ci, where U(ci)=(log(ci)−1)ci is the corresponding entropy, and (2) using a discontinuous Galerkin (DG) discretization in time. The entropy variable formulation, which was originally developed by Metti et al. (2016) under the name of a log-density formulation, guarantees both positivity of densities ci=exp(ui)>0 and a continuous-in-time energy stability result. The DG in time discretization further ensures an unconditional energy stability in the fully discrete level for any approximation order, where the lowest order case is exactly the backward Euler discretization and in this case we recover the method of Metti et al. (2016).

Convergence and Sample Complexity of Gradient Methods for the Model-Free Linear–Quadratic Regulator Problem

Model-free reinforcement learning attempts to find an optimal control action for an unknown dynamical system by directly searching over the parameter space of controllers. The convergence behavior and statistical properties of these approaches are often poorly understood because of the nonconvex nature of the underlying optimization problems and the lack of exact gradient computation. In this article, we take a step toward demystifying the performance and efficiency of such methods by focusing on the standard infinite-horizon linear–quadratic regulator problem for continuous-time systems with unknown state-space parameters. We establish exponential stability for the ordinary differential equation (ODE) that governs the gradient-flow dynamics over the set of stabilizing feedback gains and show that a similar result holds for the gradient descent method that arises from the forward Euler discretization of the corresponding ODE. We also provide theoretical bounds on the convergence rate and sample complexity of the random search method with two-point gradient estimates. We prove that the required simulation time for achieving <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -accuracy in the model-free setup and the total number of function evaluations both scale as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\log \, (1/\epsilon)$</tex-math></inline-formula> .

A splitting method for SDEs with locally Lipschitz drift: Illustration on the FitzHugh-Nagumo model

In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequencies and phases of oscillations, even for large time steps. Our results are illustrated on the stochastic FitzHugh-Nagumo model and compared with known mean-square convergent tamed/truncated variants of the Euler-Maruyama method. The capability of the proposed splitting method to preserve the aforementioned properties may make it applicable within different statistical inference procedures. In contrast, known Euler-Maruyama type methods commonly fail in preserving such properties, yielding ill-conditioned likelihood-based estimation tools or computationally infeasible simulation-based inference algorithms.

Valuation of Standard Call Options Using the Euler–Maruyama Method with Strong Approximation

Valuation of Standard Call Options Using the Euler–Maruyama Method with Strong Approximation

Solving phase-field models in the tensor train format to generate microstructures of bicontinuous composites

Phase-field models permit accounting for the underlying physics of the microstructure evolution process when simulating the emergent microstructure of a variety of multiscale materials. As an inherent characteristic, the individual phases (or states) of the material are not known in advance and only emerge upon evolution. To account for this matter, structured grids are typically used when solving phase-field models. In particular for large three-dimensional grids and long time scales, the used memory and computational time can be considerable, instigating research on alternative approaches.The work at hand studies the Tensor Train (TT) format as a possible remedy for phase-field simulations. More precisely, the TT format is a specific tensor format which permits – small so-called TT rank provided – to store huge arrays of data and comes with specific algorithms replacing the linear algebra operations familiar for fully stored arrays.We investigate the Cahn-Hilliard equation and use a semi-implicit discretization in time which is based on solving a single linear system per time step. We compare this strategy with a classic implicit Euler discretization using a Newton scheme as a nonlinear solver. Upon a finite-difference discretization in space, the resulting equations are recast in the Quantics Tensor Train (QTT) format and solved by dedicated linear solvers. We study the performance of the algorithms and investigate the effective properties of the resulting bicontinuous composites via computational homogenization. We show that the Cahn-Hilliard equation can be solved using the TT format and that runtimes scale well with the grid size.

Fly Out the Window: Exploiting Discrete-Time Flatness for Fast Vision-Based Multirotor Flight

Recent work has demonstrated fast, agile flight using only vision as a position sensor and no GPS. Current feedback controllers for fast vision-based flight typically rely on a full-state estimate, including position, velocity and acceleration. An accurate full-state estimate is often challenging to obtain due to noisy IMU measurements, infrequent position updates from the vision system, and an imperfect motion model used to obtain high-rate state estimates required by the controller. In this letter, we present an alternative control design that bypasses the need for a full-state estimate by exploiting discrete-time flatness, a structural property of the underlying vehicle dynamics. First, we show that the Euler discretization of the multirotor dynamics is discrete-time flat. This allows us to design a predictive controller using only a window of inputs and outputs, the latter consisting of position and yaw estimates. We highlight in simulation that our approach outperforms controllers that rely on an incorrect full-state estimate. We perform extensive outdoor multirotor flight experiments and demonstrate reliable vision-based navigation. In these experiments, our discrete-time flatness-based controller achieves speeds up to 10 m/s and significantly outperforms similar controllers that hinge on full-state estimation, achieving up to 80&#x0025; path error reduction.

Bulk–surface Lie splitting for parabolic problems with dynamic boundary conditions

AbstractThis paper studies bulk–surface splitting methods of first order for (semilinear) parabolic partial differential equations with dynamic boundary conditions. The proposed Lie splitting scheme is based on a reformulation of the problem as a coupled partial differential–algebraic equation system, i.e., the boundary conditions are considered as a second dynamic equation that is coupled to the bulk problem. The splitting approach is combined with bulk–surface finite elements and an implicit Euler discretization of the two subsystems. We prove first-order convergence of the resulting fully discrete scheme in the presence of a weak CFL condition of the form $\tau \leqslant c h$ for some constant $c&amp;gt;0$. The convergence is also illustrated numerically using dynamic boundary conditions of Allen–Cahn type.

Convergence rate of the EM algorithm for SDEs with low regular drifts

AbstractIn this paper we employ a Gaussian-type heat kernel estimate to establish Krylov’s estimate and Khasminskii’s estimate for the Euler–Maruyama (EM) algorithm. For applications, by taking Zvonkin’s transformation into account, we investigate the convergence rate of the EM algorithm for a class of multidimensional stochastic differential equations (SDEs) with low regular drifts, which need not be piecewise Lipschitz.

On the Lagrangian-Eulerian coupling in the immersed finite element/difference method

The immersed boundary (IB) method is a non-body conforming approach to fluid-structure interaction (FSI) that uses an Eulerian description of the momentum, viscosity, and incompressibility of a coupled fluid-structure system and a Lagrangian description of the deformations, stresses, and resultant forces of the immersed structure. Integral transforms with Dirac delta function kernels couple the Eulerian and Lagrangian variables, and in practice, discretizations of these integral transforms use regularized delta function kernels. Many different kernel functions have been proposed, but prior numerical work investigating the impact of the choice of kernel function on the accuracy of the methodology has often been limited to simplified test cases or Stokes flow conditions that may not reflect the method's performance in applications, particularly at intermediate-to-high Reynolds numbers, or under different loading conditions. This work systematically studies the effect of the choice of regularized delta function in several fluid-structure interaction benchmark tests using the immersed finite element/difference (IFED) method, which is an extension of the IB method that uses a finite element structural discretization combined with a Cartesian grid finite difference method for the incompressible Navier-Stokes equations. Whereas the conventional IB method spreads forces from the nodes of the structural mesh and interpolates velocities to those nodes, the IFED formulation evaluates the regularized delta function on a collection of interaction points that can be chosen to be denser than the nodes of the Lagrangian mesh. This opens the possibility of using structural discretizations with wide node spacings that would produce gaps in the Eulerian force in nodally coupled schemes (e.g., if the node spacing is comparable to or broader than the support of the regularized delta function). Earlier work with this methodology suggested that such coarse structural meshes can yield improved accuracy for shear-dominated cases and, further, found that accuracy improves when the structural mesh spacing is increased. However, these results were limited to simple test cases that did not include substantial pressure loading on the structure. This study investigates the effect of varying the relative mesh widths of the Lagrangian and Eulerian discretizations in a broader range of tests. Our results indicate that kernels satisfying a commonly imposed even–odd condition require higher resolution to achieve similar accuracy as kernels that do not satisfy this condition. We also find that narrower kernels are more robust, in the sense that they yield results that are less sensitive to relative changes in the Eulerian and Lagrangian mesh spacings, and that structural meshes that are substantially coarser than the Cartesian grid can yield high accuracy for shear-dominated cases but not for cases with large normal forces. We verify our results in a large-scale FSI model of a bovine pericardial bioprosthetic heart valve in a pulse duplicator.

Development of a fractional Wiener-Hermite expansion for analyzing the fractional stochastic models

The fractional Brownian motion (FBM) is a common model for long and short-range dependent phenomena that appears in different fields, including physics, biology, and finance. In the current work, a new spectral technique named the fractional Wiener Hermite Expansion (FWHE) is developed to analyze stochastic models with FBM. The technique has a theoretical background in the literature and proof of convergence. A new complete orthogonal Hermite basis set is developed. Calculus derivations and statistical analysis are performed to handle the mixed multi-dimensional fractional and/or integer-order integrals that appear in the analysis. Formulas for the mean and variance are deduced and are found to be based on fractional integrals. Using the developed expansion with the statistical properties of the basis functionals will help to reduce the stochastic model to equivalent deterministic fractional models that can be analyzed numerically or analytically with the well-known techniques. A numerical algorithm is developed to be used in case there is no available analytical solution. The numerical algorithm is compared with the fractional Euler-Maruyama (EM) technique to verify the results. In comparison to sampling based techniques, FWHE provides an efficient analytical or numerical alternative. The applicability of FWHE is demonstrated by solving different examples with additive and multiplicative FBM.

Strong convergence of Euler–Maruyama schemes for McKean–Vlasov stochastic differential equations under local Lipschitz conditions of state variables

AbstractThis paper develops strong convergence of the Euler–Maruyama (EM) schemes for approximating McKean–Vlasov stochastic differential equations (SDEs). In contrast to the existing work, a novel feature is the use of a much weaker condition—local Lipschitzian in the state variable, but under uniform linear growth assumption. To obtain the desired approximation, the paper first establishes the existence and uniqueness of solutions of the original McKean–Vlasov SDE using a Euler-like sequence of interpolations and partition of the sample space. Then, the paper returns to the analysis of the EM scheme for approximating solutions of McKean–Vlasov SDEs. A strong convergence theorem is established. Moreover, the convergence rates under global conditions are obtained.

Exponential stability of highly nonlinear hybrid NSDEs with multiple time-Dependent delays and different structures and the Euler-Maruyama method

This study focuses on the highly nonlinear hybrid neutral stochastic differential equations (NSDEs) with multiple time-dependent delays and different structures. The new stochastic system in this paper contains multiple time-dependent delays in the neutral term, and the coefficients are also with different delay functions in drift and diffusion terms respectively. Besides, the set of Markov switching states are divided into two subsets, and the structures of the system are different in each of them. Firstly, the existence, uniqueness and asymptotic boundedness of the global solution of the new NSDE are proved, and then the novel criteria of pth moment and almost surely exponential stability are investigated. All these results are obtained under highly nonlinear coefficients that satisfy local Lipschitz condition and Khasminskii-type conditions, and the Lyapunov function method and the generalized Itô formula are mainly used. Moreover, the Euler-Maruyama approximate solution is established and proved to be convergent in probability to the theoretical solution. Finally, a numerical example is presented together with the corresponding computer simulation to demonstrate the effectiveness of the main results.

An Improved Deadbeat Current Controller of PMSM Based on Bilinear Discretization

Based on the bilinear discretization mathematical model of permanent magnet synchronous motor (PMSM), an improved incremental deadbeat current prediction control algorithm is proposed. Aiming at the system instability caused by the forward Euler discretization method, this paper combines the deadbeat current prediction control and the improved bilinear discretization method to improve the system stability. Further, the proposed controller considers the two-beat delay of a digital system to make the mathematical model more accurate. Moreover, the proposed bilinear discretization predictive current controller is not affected by the permanent magnet flux of the motor. Then, the system stability conditions of the proposed controller are analyzed. The simulation and experimental results verify the feasibility and effectiveness of the proposed method.

Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation

&lt;p style='text-indent:20px;'&gt;This paper considers the initial value problem of general nonlinear stochastic fractional integro-differential equations with weakly singular kernels. Our effort is devoted to establishing some fine estimates to include all the cases of Abel-type singular kernels. Firstly, the existence, uniqueness and continuous dependence on the initial value of the true solution under local Lipschitz condition and linear growth condition are derived in detail. Secondly, the Euler–Maruyama method is developed for solving numerically the equation, and then its strong convergence is proven under the same conditions as the well-posedness. Moreover, we obtain the accurate convergence rate of this method under global Lipschitz condition and linear growth condition. In particular, the Euler–Maruyama method can reach strong first-order superconvergence when &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \alpha = 1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. Finally, several numerical tests are reported for verification of the theoretical findings.&lt;/p&gt;

Stability of Cyber-Physical Systems of Numerical Methods for Stochastic Differential Equations: Integrating the Cyber and the Physical of Stochastic Systems

This paper presents the cyber-physical system (CPS) of a numerical method (the widely-used Euler- Maruyama method) and establishes a foundational theory of the CPSs of numerical methods for stochastic differential equations (SDEs), which transforms the way we understand the relationship between the numerical method and the underlying dynamical system. Unlike in the literature where they are treated as separate systems linked by inequalities, the CPS is a seamless integration of the SDE and the numerical method and we construct a new and general class of stochastic impulsive differential equations (SiDEs) that can serve as a canonic form of the CPSs of numerical methods. By the CPS approach, we show the equivalence and intrinsic relationship between the stability of the SDE and the stability of the numerical method using the Lyapunov stability theory we develop for our class of SiDEs. Applying our established theory, we present the CPS Lyapunov inequality that is the necessary and sufficient condition for meansquare stability of the CPS of the Euler-Maruyama method for linear SDEs. The proposed CPS and theory initiate the study of systems numerics and provoke many open and interesting problems for future work.

Strong convergence of the Euler-Maruyama method for the generalized stochastic Volterra integral equations driven by Lévy noise

In this paper, the theoretical and numerical analysis of the stochastic Volterra integral equations (SVIEs) driven by L?vy noise are considered. We investigate the existence, uniqueness, boundedness and H?lder continuity of the analytic solutions for SVIEs driven by L?vy noise. The Euler-Maruyama method for SVIEs driven by L?vy noise is proposed. The boundedness of the numerical solution is proved, and the strong convergence order is obtained. Some numerical examples are given to support the theoretical results.