A Note on Maximal Inequality for Stochastic Convolutions

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Well Posedness and Asymptotic Behavior for Stochastic Reaction-Diffusion Equations with Multiplicative Poisson Noise (2009, methodology): “Moreover, the stochastic convolution also admits a càdlàg modification by the dilation method, as in [17] or [31, p.”
On Maximal Inequalities for Purely Discontinuous Martingales in Infinite Dimensions (2013, background): “The idea of using dilation theorems to extend results from stochastic integrals to stochastic convolutions has been introduced, to the best of our knowledge, in [12].”
Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise (2008, unknown): “We shall follow the approach of [14].”
LOCAL WELL‐POSEDNESS OF MUSIELA’S SPDE WITH LÉVY NOISE (2007, unknown): “We shall follow the approach of [10].”
Stochastic Convolutions Driven by Martingales: Maximal Inequalities and Exponential Integrability (2007, methodology): “We used the “unitary dilations trick” in our previous article [22] to study stochastic convolutions driven by Wiener processes.”