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

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In this article we obtain rates of convergence to equilibrium of marked Hawkes processes in two situations. Firstly, the stationary process is the empty process, in which case we speak of the rate of extinction. Secondly, the stationary process is the unique stationary and nontrivial marked Hawkes process, in which case we speak of the rate of installation. The first situation models small epidemics, whereas the results in the second case are useful in deriving stopping rules for simulation algorithms of Hawkes processes with random marks.

A. CALZOLARI" and G. NAPPO" "L)rpnrtm~eizto dr Maf~nzatrcn-l!niv~ri.rra rir Roma "Tor Pergata -krcr della Riceuccr Scic~ificu-OOI33 Romu-Italia; b~ipartin-2twfn di Mat~matica-Universitd diRorna "La Sapienza' LPiazzale A.Muru. 2-00185 Roma-ItuliaLet (X. Y ) be a pure jump Markov process with discrete state space. Let the state X be not observable and the observation Y be a counting process. We are interested in the filter of X given Y and in its dependence on the model. More precisely we compare this filter with the filter of another system which differs from the previous one only by the infinitesimal parameters and the initial distribution, and we give an explicit bound for the distance in variation norm between the two filters. Finally we use this bound to examine how much a discrete time approximation procedure is affected by a slight error In the model and, in a special case, to examine the error due to the use of a finite state space model instead of an infinitc onc.K~y w o r k : Filtering; robustness; counting process: jump Markov process; couplingIn non linear filtering theory two problems naturally arise. The first one is due to the difficulty of computing explicitely the filter and leads to approximation schemes (See [3] as a basic survey on this subject). The second one is due to uncertainty on the model and leads to investigate the sensitivity (robustness) of the filter with respect to the use of models that differ, for instance, in some parameters and/or in the initial distributions. Many authors deal with these problems and we may quote [8] as the first contribution on the combined problem of robust approximation. We recall also [7], where, for a rather general diffusion model, the uncertainty of the model is treated from a Bayesian point of view and an error bound is given. Stochastics and Stochastic Reports 1997.61:1-19. Y = ( ( ) = , A ( A ' and / ihai qL-7-j i h ' l " ! i j -eso ii--j and Ejc7-j < Ei. !vs!A! 3j J -' C J p JiL-J= exp {Xi(c -1)T). By taking expectations w.r.t. P and in (2.7) and by using the above bounds, we get (2.8).

We study some approximation schemes for a nonlinear filtering problem when the state process X is the solution of a stochastic delay diffusion equation, and the observation process is a noisy function of X s for s ∈ [t − τ, t], where τ is a constant. The approximating state is given by means of an Euler discretization scheme, and the observation process is a noisy function of the approximating state.