On the Moments of the Modulus of Continuity of Ito Processes

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 Ito processes with possibly unbounded coefficients, starting from the special case of Brownian motion. References to known results for the case of Brownian motion and Ito 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.