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

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 $p\ge 2$ and apply the result to obtain confidence interval estimations for the approximate solutions.