On Émery’s inequality and a variation-of-constants formula

A generalization of Émery’s inequality for stochastic integrals is shown for convolution integrals of the form ( ∫ t 0 g(t − s)Y (s−) dZ(s))t>0, where Z is a semimartingale, Y an adapted càdlàg process, and g a deterministic function. The function g is assumed to be absolutely continuous with a derivative that is continuous or of bounded variation or a sum of such functions. The function g may also have jumps, as long as the jump sizes are absolutely summable. The inequality is used to prove existence and uniqueness of solutions of equations of variation-of-constants type. As a consequence, it is shown that the solution of a stochastic delay differential equation with linear drift, bounded functional Lipschitz diffusion coefficient, and noise driven by a general semimartingale satisfies a variation-of-constants formula. The proof of the inequality consists of an approximation argument and a reduction to Émery’s original inequality.