On the Moments of the Modulus of Continuity of Itô Processes

Abstract

The modulus of continuity of a stochastic process is a random element for any fixed mesh size. We provide upper bounds for the moments of the modulus of continuity of Itô processes with possibly unbounded coefficients, starting from the special case of Brownian motion. References to known results for the case of Brownian motion and Itô processes with uniformly bounded coefficients are included. As an application, we obtain the rate of strong convergence of Euler–Maruyama schemes for the approximation of stochastic delay differential equations satisfying a Lipschitz condition in supremum norm.

Keywords:

• Delay
• Euler–Maruyama scheme
• Extreme values
• Functional differential equation
• Itô process
• Modulus of continuity
• Stochastic differential equation

Mathematics Subject Classification:

• Primary 60J60, 60J65, 60G17, 60G70, 60H99
• Secondary 34K50, 60H10, 60H35

Acknowledgments

Financial support from the following projects of the University of Rome “La Sapienza” is gratefully acknowledged: “Processi stocastici e applicazioni” (2005 – prot. C26A058977), “Strutture di dipendenza in modelli stocastici e applicazioni” (2007 – prot. C26A07HZZJ).