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

TY - JOUR

T1 - On Émery's Inequality and a Variation-of-Constants Formula

AU - Riedle, Markus

AU - Reiß, Markus

AU - van Gaans, Onno

PY - 2007

Y1 - 2007

N2 - A generalization of Émery's inequality for stochastic integrals is shown for convolution integrals of the form ( \int_0^t g(t-s) Y(s-) dZ(s))_{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.

AB - A generalization of Émery's inequality for stochastic integrals is shown for convolution integrals of the form ( \int_0^t g(t-s) Y(s-) dZ(s))_{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.

KW - Émery’s inequality;

KW - Functional Lipschitz coefficient;

KW - Linear drift;

KW - Semimartingale

KW - Stochastic delay differential equation;

KW - Variation-of-constants formula

U2 - 10.1080/07362990601139586

DO - 10.1080/07362990601139586

M3 - Article

SN - 0736-2994

VL - 25

SP - 353

EP - 379

JO - STOCHASTIC ANALYSIS AND APPLICATIONS

JF - STOCHASTIC ANALYSIS AND APPLICATIONS

IS - 2

ER -