Let $Z$ be an increasing Levy process (i.e. a subordinator). Let $\lambda>0$ and consider the Ornstein Uhlenbeck type SDE $$ d V_t = - \lambda V_t dt + d Z_{\lambda t } $$ where the integral can e.g. be defined using Lebesgue-Stieltjes. The solution can e.g. be written $$ V_T = V_0 e^{-\lambda T} + \int_0^T e^{-\lambda(T-s)} dZ_{\lambda s}. $$
Now let $t,s\geq 0$ can one then show that there exists $K>0$ such that $$ E[\lvert V_t - V_s \rvert^2 \wedge K] \leq K\lvert t-s \rvert? $$ It is an assumption I need satisfied for an application. I've tried to get the difference expressed as a difference of Levy processes to use the stationary, independent increments but I keep ending up with time intervals multiplied. Any ideas? What type of assumption is this anyway?
The source claims it is true "when the evolution of $V_t$ is given by a (multivariate) stochastic differential equation which is a common modeling approach for jump-diffusive asset returns displaying stochastic volatility. "
So a reference for this might suffice.