Convergence analysis of semi-implicit Euler methods for solving stochastic equations with variable delays and random jump magnitudes (Q629527)

A semi-implicit Euler method (SIE) is presented for solving systems of stochastic differential equations with variable delays and random jump magnitudes of the form \[ \begin{multlined} dX(t)= f(X(t), X(t-\tau(t)))\,dt+ g(X(t), X(t-\tau(t)))\,dW^t)+\\ h(X(t),X(t-\tau(t)),\gamma_{N(t)+ 1})\,dN(t)\end{multlined} \] with \(0\leq t\leq T\) and \(X(t)= \psi(t)\), \(-r\leq t\leq 0\), where \(\tau(t)\) is a variable delay, \(W\) is a standard \(m\)-dimensional Wiener process, \(N(t)\) is a Poisson process, and \(\gamma_i\), \(i=1,2,\dots\) are independent identically distributed random variables. It is proved that the SIE approximate solutions converge to the exact solution of the system both in the mean-square and in probability.