# On Émery's Inequality and a Variation-of-Constants Formula

Reiß, Markus and Riedle, Markus and van Gaans, Onno (2007) On Émery's Inequality and a Variation-of-Constants Formula. Stochastic Analysis and Applications, 25 (2). pp. 353-379. ISSN 1532-9356

A generalization of Émery's inequality for stochastic integrals is shown for convolution integrals of the form $\left( \int_0^t g(t-s) Y(s-) dZ(s)\right)_{t \geq 0}$, where Z is a semimartingale, Y an adapted càdlàg process, and g a deterministic function. An even more general inequality for processes with two parameters is proved. 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 semilinear delay differential equation with functional Lipschitz diffusion coefficient and driven by a general semimartingale satisfies a variation-of-constants formula.