Convergence analysis of semi-implicit Euler methods for solving stochastic equations with variable delays and random Jump magnitudes

Abstract

In this paper, we study the following stochastic equations with variable delays and random jump magnitudes: { d X ( t ) = f ( X ( t ) , X ( t − τ ( t ) ) ) d t + g ( X ( t ) , X ( t − τ ( t ) ) ) d W ( t ) + h ( X ( t ) , X ( t − τ ( t ) ) , γ N ( t ) + 1 ) d N ( t ) , 0 ≤ t ≤ T , X ( t ) = ψ ( t ) , − r ≤ t ≤ 0 . We establish the semi-implicit Euler approximate solutions for the above systems and prove that the approximate solutions converge to the analytical solutions in the mean-square sense as well as in the probability sense. Some known results are generalized and improved.