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

• Published in: Mathematical and computer modelling Vol. 54; no. 9; pp. 2235 - 2251
• Main Author: MILOSEVIC, Marija
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01.11.2011; Elsevier
• Subjects: Convergence in probability; equations; Euler–Maruyama method; Exact sciences and technology; Khasminskii-type conditions; Mathematical analysis; Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); Neutral stochastic differential equations; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in probability and statistics; probability; Sciences and techniques of general use; Sequences, series, summability; Time-dependent delay; Euler–Maruyama method; Neutral stochastic differential equations; Khasminskii-type conditions; Time-dependent delay; Convergence in probability; Solution uniqueness; Euler-Maruyama method; Existence condition; Differential equation; Probability distribution; Delay equation; Euler scheme; Discrete scheme; Convergence; Exact solution; Non linear equation; Neutral equation; Applied mathematics; Discrete time; Integral equation; Mathematical model; Computer aided analysis; Stochastic equation
• ISSN: 0895-7177; 1872-9479
• DOI: 10.1016/j.mcm.2011.05.033

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.