Marija Milošević

Almost sure exponential stability of solutions to highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler–Maruyama approximation

Abstract This paper may be considered as a natural sequel to the paper [M. Milosevic, Highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler–Maruyama method, Mathematical and Computer Modelling 54 (2011) 2235–2251]. In the present paper, global almost sure (a.s.) asymptotic exponential stability of the equilibrium solution for a class of neutral stochastic differential equations with time-dependent delay is considered, under nonlinear growth conditions. Additionally, the moment estimates are established for solutions of equations of this type. Under more restrictive conditions, including the linear growth condition, we show that the appropriate Euler–Maruyama equilibrium solution is globally a.s. asymptotically exponentially stable. As expected, the whole consideration is affected by the presence and properties of the delay function. In that sense, the delayed terms are explicitly treated as arguments of the coefficients of the equation and, particularly, under the derivative of the state variable. Additionally, some requirements related to the rate of change of the delay function are imposed in order to provide the main results of the paper.

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

Abstract 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.

A Taylor polynomial approach in approximations of solution to pantograph stochastic differential equations with Markovian switching

The subject of this paper are analytic approximate methods for pantograph stochastic differential equations with Markovian switching, as well as their counterparts without Markovian switching. Approximate equations are defined on equidistant partitions of the time interval, and their coefficients are Taylor approximations of the coefficients of the initial equation. In the case with Markovian switching we will present the approximate method based on Taylor approximation of coefficients in two arguments and show that the appropriate approximate solutions converge in the L^p-norm to the solution of the initial equation. Then we will present the other approximate method which deals with Taylor approximation in the first argument. In both cases the closeness between the approximate solution and the solution of the initial equation depends on the number of degrees in Taylor approximations of coefficients, although the presence of the Markov chain affects it. These approximate methods are then adapted to the case without Markovian switching. The first method gives the possibility of proving L^p convergence as well as a.s. convergence of the appropriate sequence of approximate solutions to the solution of the initial equation.

Existence, uniqueness, almost sure polynomial stability of solution to a class of highly nonlinear pantograph stochastic differential equations and the Euler–Maruyama approximation

Abstract In the first part of this paper the existence, uniqueness and almost sure polynomial stability of solutions for pantograph stochastic differential equations are considered, under nonlinear growth conditions. The results are obtained using the idea from (Mao and Rassias, 2005) [10], where stochastic differential equation with constant delay are considered. However, the presence of the unbounded delay in stochastic pantograph differential equations required certain modification of that idea. Moreover, the convergence in probability of the appropriate Euler–Maruyama solution is proved under the same nonlinear growth conditions. Adding the linear growth condition, we show that the almost sure polynomial stability of the Euler–Maruyama solution implies the almost sure polynomial stability of the exact solution. This part of the paper represents the extension of the idea from (Wu et al., 2010) [17]. The main novelty in this part of the paper is also related to the treatment of the unbounded delay in pantograph stochastic differential equations.

Stability of a class of neutral stochastic differential equations with unbounded delay and Markovian switching and the Euler-Maruyama method

This paper is motivated by the paper Hu etźal. (2013). This paper contains the existence and uniqueness, as well as stability results of the exact solution for a class of neutral stochastic differential equations with unbounded delay and Markovian switching, including the case when the delay function is bounded. Moreover, the convergence in probability of the Euler-Maruyama method is established regardless whether or not the delay function is bounded. These results are obtained under certain non-linear growth conditions on the coefficients of the equation. Adding the linear growth condition on the drift coefficient, the almost sure exponential stability of the Euler-Maruyama method is proved in the case of the bounded delay. The presence of the neutral term is essential for consideration of this class of equations. It should be stressed that the neutral term is also hybrid, that is, it depends on the Markov chain. Moreover, the Euler-Maruyama method is defined in a non-trivial way regarding the neutral term.

An application of Taylor series in the approximation of solutions to stochastic differential equations with time-dependent delay

The subject of this paper is the analytic approximation method for solving stochastic differential equations with time-dependent delay. Approximate equations are defined on equidistant partitions of the time interval, and their coefficients are Taylor approximations of the coefficients of the initial equation. It will be shown, without making any restrictive assumption for the delay function, that the approximate solutions converge in L^p-norm and with probability 1 to the solution of the initial equation. Also, the rate of the L^p convergence increases when the degrees in the Taylor approximations increase, analogously to what is found in real analysis. At the end, a procedure will be presented which allows the application of this method, with the assumption of continuity of the delay function.

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

Abstract 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.

Implicit numerical methods for highly nonlinear neutral stochastic differential equations with time-dependent delay

Abstract This paper represents the continuation of the analysis from papers Milosevic (2011) [10] and Milosevic (2013) [11]. The main aim of this paper is to establish certain results for the backward Euler method for a class of neutral stochastic differential equations with time-dependent delay. For that purpose, the split-step backward Euler method, which represents an extension of the backward Euler method, is introduced for this class of equations. Conditions under which the split-step backward Euler method, and thus the backward Euler method, is well defined are revealed. Moreover, the convergence in probability of the backward Euler method is proved under certain nonlinear growth conditions including the one-sided Lipschitz condition. This result is proved using the technique which is based on the application of the continuous-time approximation. For this reason, the discrete forward–backward Euler method is involved since it allows its continuous version to be well defined from the aspect of measurability. The convergence in probability is established for the continuous forward–backward Euler solution, which is essential for proving the same result for both discrete forward–backward and backward Euler methods. Additionally, it is proved that the discrete backward Euler equilibrium solution is globally a.s. asymptotically exponentially stable, without the linear growth condition on the drift coefficient of the equation. As usual, the whole consideration is affected by the presence and properties of the delay function.

Convergence and almost sure exponential stability of implicit numerical methods for a class of highly nonlinear neutral stochastic differential equations with constant delay

This paper can be regarded as the continuation of the work contained in papers Milosevic (2011, 2013). At the same time, it represents the extension of the paper Wu et?al. (2010). In this paper, the one-sided Lipschitz condition is employed in the context of the backward Euler method, for a class of neutral stochastic differential equations with constant delay. Sufficient conditions for this method to be well defined are revealed. Under certain nonlinear growth conditions, the convergence in probability is established for the continuous forward-backward Euler method, as well as for the discrete backward Euler method. Additionally, it is proved that the discrete backward Euler equilibrium solution is globally a.s. asymptotically exponentially stable, without requiring for the drift coefficient to satisfy the linear growth condition. This manuscript represents the continuation of the previous work of the author, related to a class of highly nonlinear neutral stochastic differential equations with time-dependent delay.Main results of this manuscript are convergence in probability and almost sure exponential stability of the backward Euler approximate solution for a class of stochastic differential equations with constant delay.Conditions under which both, the exact and approximate solutions share the property of the almost sure exponential stability are revealed.This paper also illustrates that, in some cases, stochastic differential equations with constant delay should be studied separately from those with time-dependent delay. In that sense, it should be stressed that results of this paper are obtained under weaker conditions comparing to those which would be obtained by replacing the time-dependent delay by the constant delay.

Convergence and almost sure polynomial stability of the backward and forward–backward Euler methods for highly nonlinear pantograph stochastic differential equations

Abstract In this paper the backward Euler and forward–backward Euler methods for a class of highly nonlinear pantograph stochastic differential equations are considered. In that sense, convergence in probability on finite time intervals is established for the continuous forward–backward Euler solution, under certain nonlinear growth conditions. Under the same conditions, convergence in probability is proved for both discrete forward–backward and backward Euler methods. Additionally, under certain more restrictive conditions, which do not include the linear growth condition on the drift coefficient of the equation, it is proved that these solutions are globally a.s. asymptotically polynomially stable. Numerical examples are provided in order to illustrate theoretical results.