Modern Stochastics: Theory and Applications

For a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset $D\subset {\mathbb{R}^{d}}$ and driven by an ${L^{2}}(D)$-valued fractional Brownian motion with the Hurst index $H>1/2$, a new result on existence and uniqueness of a mild solution is established. Compared to the existing results, the uniqueness in a fully nonlinear case is shown, not assuming the coefficient in front of the noise to be affine. Additionally, the existence of moments for the solution is established.

A one-dimensional stochastic wave equation driven by a general stochastic measure is studied in this paper. The Fourier series expansion of stochastic measures is considered. It is proved that changing the integrator by the corresponding partial sums or by Fejèr sums we obtain the approximations of mild solution of the equation.

A stochastic heat equation on $[0,T]\times \mathbb{R}$ driven by a general stochastic measure $d\mu (t)$ is investigated in this paper. For the integrator μ, we assume the σ-additivity in probability only. The existence, uniqueness, and Hölder regularity of the solution are proved.