Euler-Maruyama Approximate Solution Research Articles

On the rate of convergence of Euler–Maruyama approximate solutions of stochastic differential equations with multiple delays and their confidence interval estimations

<abstract><p>In this paper, we investigate Euler–Maruyama approximate solutions of stochastic differential equations (SDEs) with multiple delay functions. Stochastic differential delay equations (SDDEs) are generalizations of SDEs. Solutions of SDDEs are influenced by both the present and past states. Because these solutions may include past information, they are not necessarily Markov processes. This makes representations of solutions complicated; therefore, approximate solutions are practical. We estimate the rate of convergence of approximate solutions of SDDEs to the exact solutions in the $ L^p $-mean for $ p \geq 2 $ and apply the result to obtain confidence interval estimations for the approximate solutions.</p></abstract>

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.

Truncated EM numerical method for generalised Ait-Sahalia-type interest rate model with delay

The original Ait-Sahalia model of the spot interest rate proposed by Ait-Sahalia assumes constant volatility. As supported by several empirical studies, volatility is never constant in most financial markets. From application viewpoint, it is important we generalise the Ait-Sahalia model to incorporate volatility as a function of delay in the spot rate. In this paper, we study analytical properties for the true solution of this model and construct several new techniques of the truncated Euler–Maruyama (EM) method to study properties of the numerical solutions under the local Lipschitz condition plus Khasminskii-type condition. Finally, we justify that the truncated EM approximate solution can be used within a Monte Carlo scheme for numerical valuations of some financial instruments such as options and bonds.

Convergence of the EM method for NSDEs with time-dependent delay in the G-framework

ABSTRACTConsider a neutral stochastic differential equation (NSDE) with time-dependent delay () in the G-framework where denotes a G-Brownian motion and the quadratic variation process of . We introduce an Euler–Maruyama (EM) method for solving this equation and prove that the EM approximate solution converges to the exact solution with a strong order of the mean square closeness equal to one under the global Lipschitz condition. A numerical example is provided to illustrate the effectiveness of our method.

Almost sure exponential stability of the $$\theta $$ θ -Euler–Maruyama method, when $$\theta \in (\frac{1}{2},1)$$ θ ∈ ( 1 2 , 1 ) , for neutral stochastic differential equations with time-dependent delay under nonlinear growth conditions

This paper is motivated by Lan and Yuan (J Comput Appl Math 285:230–242, 2015). The main aim of this paper is to establish certain results for the $$\theta $$ -Euler–Maruyama method for a class of neutral stochastic differential equations with time-dependent delay. The method is defined such that, in general case, it is implicit in both drift and neutral term. The drift and neutral term are both parameterized by $$\theta $$ in a way which guarantees that for $$\theta =0$$ and $$\theta =1$$ , the method reduces to the Euler–Maruyama method and backward Euler method, respectively, which can be found in the literature. The one-sided Lipschitz conditions in the present-state and delayed arguments of the drift coefficient of this class of equations for any $$\theta \in [0,1]$$ are employed in order to guarantee the existence and uniqueness of the appropriate $$\theta $$ -Euler–Maruyama approximate solution. The main result of this paper is almost sure exponential stability of the $$\theta $$ -Euler–Maruyama method, for $$\theta \in (\frac{1}{2},1),$$ under nonlinear growth conditions. Some comments and conclusions are presented for the corresponding deterministic case. An example and numerical simulations are provided to support the main results of the paper.

Interval estimations for Euler-Maruyama approximate solutions of stochastic differential equations with multiple delays

Interval estimations for Euler-Maruyama approximate solutions of stochastic differential equations with multiple delays

Estimates for the difference between approximate and exact solutions to stochastic differential equations in the G-framework

ABSTRACTThis article investigates the Euler-Maruyama approximation procedure for stochastic differential equations in the framework of G-Browinian motion with non-linear growth and non-Lipschitz conditions. The results are derived by using the Burkholder-Davis-Gundy (in short BDG), Hölder's, Doobs martingale's and Gronwall's inequalities. Subject to non-linear growth condition, it is revealed that the Euler-Maruyama approximate solutions are bounded in . In view of non-linear growth and non-uniform Lipschitz conditions, we give estimates for the difference between the exact solution and approximate solutions of SDEs in the framework of G-Brownian motion.

Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation

<p style='text-indent:20px;'>In this paper, we are concerned with the asymptotic properties and numerical analysis of the solution to hybrid stochastic differential equations with jumps. Applying the theory of M-matrices, we will study the <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>th moment asymptotic boundedness and stability of the solution. Under the non-linear growth condition, we also show the convergence in probability of the Euler-Maruyama approximate solution to the true solution. Finally, some examples are provided to illustrate our new results.

On existence and approximate solutions for stochastic differential equations in the framework of G-Brownian motion

This investigation aims at studying a Euler-Maruyama (EM) approximate solutions scheme for stochastic differential equations (SDEs) in the framework of G-Brownian motion. Subject to the growth condition, it is shown that the EM solutions $ Z^q(t)$ are bounded, in particular, $ Z^q(t)\in M_G^2([t_0,T];\mathbb{R}^n)$ . Letting Z(t) as a unique solution to SDEs in the G-framework and utilizing the growth and Lipschitz conditions, the convergence of $ Z^q(t)$ to Z(t) is revealed. The Burkholder-Davis-Gundy (BDG) inequalities, Holder’s inequality, Gronwall’s inequality and Doobs martingale’s inequality are used to derive the results. In addition, without assuming a solution of the stated SDE, we have shown that the Euler-Maruyama approximation sequence $ \{Z^q(t)\}$ is Cauchy in $ M_G^2([t_0,T];\mathbb{R}^n)$ thus converges to a limit which is a unique solution to SDE in the G-framework.

Strong convergence of implicit numerical methods for nonlinear stochastic functional differential equations

Abstract The main aim of this work is to prove that the backward Euler–Maruyama approximate solutions converge strongly to the true solutions for stochastic functional differential equations with superlinear growth coefficients. The paper also gives the boundedness and mean-square exponential stability of the exact solutions, and shows that the backward Euler–Maruyama method can preserve the boundedness of mean-square moments. Finally, a highly nonlinear example is provided to illustrate the main results.

Numerical approximation of stochastic differential delay equation with coefficients of polynomial growth

Although numerical methods of nonlinear stochastic differential delay equations (SDDEs) have been discussed by several authors, there is so far little work on the numerical approximation of SDDE with coefficients of polynomial growth. The main aim of the paper is to investigate convergence in probability of the Euler-Maruyama (EM) approximate solution for SDDE with one-sided polynomial growing drift coefficient and polynomial growing diffusion coefficient. Moreover, we prove the existence-and-uniqueness of almost surely exponentially stable global solution for this nonlinear stochastic delay system. Finally, a computer simulation confirms the efficiency of our numerical method.

The Euler–Maruyama approximation of solutions to stochastic differential equations with piecewise constant arguments

In this paper we consider stochastic differential equations with piecewise constant arguments. These equations describe hybrid dynamical systems, that is, combinations of continuous and discrete systems. Our aim is to establish the mean square convergence of the Euler–Maruyama approximate solution under quite general conditions, that is, the global Lipschitz condition and the linear growth condition, which guarantee the existence and uniqueness of the true solution. Then we show that the initial equation is exponentially stable in mean square if and only if, for some sufficiently small step-size Δ, the Euler–Maruyama method is exponentially stable in mean square. The stability study does not involve the Lyapunov functions nor functionals. It should be pointed out that stochastic differential equations with piecewise constant arguments can be regarded as a class of stochastic differential equations with multiple time-dependent delays which can be found in the literature. However, in the convergence analysis of the Euler–Maruyama method of these equations, it is required that the delay functions satisfy Lipschitz continuity condition. In the present paper, the delay functions do not satisfy that condition, so it represents an extension of results from the papers Mao (2003, 2007), using the technique similar to that from the first paper.

Confidence Intervals for the Euler-Maruyama Approximate Solutions of Stochastic Delay Differential Equations

Confidence Intervals for the Euler-Maruyama Approximate Solutions of Stochastic Delay Differential Equations

A Confidence Interval for the Euler-Maruyama Approximate Solution of SDE with Unbounded Coefficients

A Confidence Interval for the Euler-Maruyama Approximate Solution of SDE with Unbounded Coefficients

Exponential stability for nonlinear hybrid stochastic pantograph equations and numerical approximation

The paper develops exponential stability of the analytic solution and convergence in probability of the numerical method for highly nonlinear hybrid stochastic pantograph equation. The classical linear growth condition is replaced by polynomial growth conditions, under which there exists a unique global solution and the solution is almost surely exponentially stable. On the basis of a series of lemmas, the paper establishes a new criterion on convergence in probability of the Euler-Maruyama approximate solution. The criterion is very general so that many highly nonlinear stochastic pantograph equations can obey these conditions. A highly nonlinear example is provided to illustrate the main theory.

Strong convergence of the stopped Euler–Maruyama method for nonlinear stochastic differential equations

In this paper, numerical methods for the nonlinear stochastic differential equations (SDEs) with non-global Lipschitz drift coefficient are discussed. The existing known results have only so far shown that the classical (explicit) Euler–Maruyama (EM) approximate solutions converge to the true solution in probability [22,23]. More recently, the authors in [16] proved that the classical EM method will diverge in L2 sense for the underlying SDEs in this paper (and those SDEs with superlinearly growing coefficients). These strongly indicate that the classical EM method is not good enough for the highly nonlinear SDEs. However, in this paper, we introduce a modified EM method using stopping time and show successfully that the discrete version of the modified EM approximate solution converges to the true solution in the strong sense (namely in L2) with a order arbitrarily close to a half.

Numerical approximation of nonlinear neutral stochastic functional differential equations

The paper investigates numerical approximations for solution of neutral stochastic functional differential equation (NSFDE) with coefficients of the polynomial growth. The main aim is to develop the convergence in probability of Euler-Maruyama approximate solution under highly nonlinear growth conditions. The paper removes the linear growth condition of the existing results replacing by highly nonlinear growth conditions, so the convergence criteria here may cover a wider class of nonlinear systems. Moreover, we also prove the existence-and-uniqueness of the global solutions for NSFDEs with coefficients of the polynomial growth. Finally, two examples is provided to illustrate the main theory.

Discrete Razumikhin-type technique and stability of the Euler--Maruyama method to stochastic functional differential equations

A discrete stochastic Razumikhin-type theorem is established to investigate whether the Euler--Maruyama (EM) scheme can reproduce the moment exponential stability of exact solutions of stochastic functional differential equations (SFDEs). In addition, the Chebyshev inequality and the Borel-Cantelli lemma are applied to show the almost sure stability of the EM approximate solutions of SFDEs. To show our idea clearly, these results are used to discuss stability of numerical solutions of two classes of special SFDEs, including stochastic delay differential equations (SDDEs) with variable delay and stochastically perturbed equations.

JUMP SYSTEMS WITH THE MEAN-REVERTING γ-PROCESS AND CONVERGENCE OF THE NUMERICAL APPROXIMATION

In this paper, a class of jump systems with the mean-reverting γ-process are considered in finance. The analytical properties including the positivity, boundedness and pathwise estimations of the solution are discussed. Moreover, the authors show that the Euler–Maruyama approximate solutions converge to the true solutions in probability. Finally, the authors apply the convergence to examine a bond and a path-dependent option price in the financial pricing.

Highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler–Maruyama method

The subject of this paper is the development of discrete-time approximations for solutions of a class of highly nonlinear neutral stochastic differential equations with time-dependent delay. The main contribution is to establish the convergence in probability of the Euler–Maruyama approximate solution without the linear growth condition, that is, under Khasminskii-type conditions. The presence of the delayed argument in the equation, especially in the derivative of the state variable, requires a special treatment and some additional conditions, except the conditions that guarantee the existence and uniqueness of the exact solution. The existence and uniqueness result and the convergence in probability are directly influenced by the properties of the delay function.