Wave equation with unbounded terms driven by a stochastic measure
The one-dimensional wave equation with additive noise driven by a stochastic measure is considered. For the stochastic integrator, we assume only the sigma-additivity in probability and that $|y|^τ$ is integrable on $\mathbb{R}$ for some $τ>1/2$. The class of stochastic measures includes many important cases of random processes. In the equation, the stochastic measure depends only on the spatial variable. The existence, uniqueness, and Hölder regularity of the mild solution are proved. In the previous publications, similar results were obtained for the case of bounded terms of the wave equation driven by a stochastic measure. In our paper, one term may be unbounded, satisfying the linear growth condition. Nevertheless, we prove that the solution is bounded in this case.