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

Abstract

A generalization ofÉmery'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 toÉmery's original inequality.

Delay differential equations driven by Lévy processes: Stationarity and Feller properties

“…This formula is easily derived if the driving process has bounded second moments, but no longer for processes where an Itô isometry or inequality fails. We provide a proof separately in Reiß et al [26].…”

Bubbles and crashes in a Black–Scholes model with delay

“…The dependence of the solutions on the initial condition ϕ is neglected in our notation in what follows; that is, we will write y(t) = y(t, ϕ) and Y (t) = Y (t, ϕ) for the solutions of (2.1) and (2.5), respectively. By Reiß et al [22,Lemma 6.1] the solution (Y (t) : t ≥ −τ ) of (2.5) obeys a variation of constants formula 6) where r is the fundamental solution of (2.1).…”

Numerical Picard Iteration Methods for Simulation of Non-Lipschitz Stochastic Differential Equations

“…For the stochastic differential equation (SDE), we define the homogeneous SDE as an linear or linearised SDE, which can be solved analytically or numerically, see [17][18][19]. Further, there exists also ideas for SDE and stochastic partial differential equations (SPDEs) to decompose into linear and nonlinear parts of the SDE, which can be solved linear and nonlinear stochastic methods, see [20].…”