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

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.