We study stochastic lattice differential equations with Hölder noise where the stochastic integral is defined pathwise in Gubinelli’s sense [27]. We prove the existence of a random absorbing set by evaluating the difference of the solution and that of the corresponding deterministic equation on each interval of a greedy sequence of stopping times. The asymptotic compactness of the generated random dynamical system is then proved. That implies the existence of a global random pullback attractor \(\mathcal {A}\) for the equation. Similar result holds true for the discretization system in the form of the explicit Euler scheme defined on a regular time set. Moreover, the attractor of the discrete system converges upper semi-continuously to \(\mathcal {A}\) as the step size tends to zero. We also consider the system in finite dimensional spaces. We show the upper semi-continuity of the random pullback attractor \(\mathcal {A}\) in the sense that the random attractors of discrete systems in finite dimensional spaces converge to that of the original lattice system.
