Convergence of stochastic integrals with applications to transport equations and conservation laws with noise

Convergence of stochastic integrals driven by Wiener processes $W_n$, with $W_n \to W$ almost surely in $C_t$, is crucial in analyzing SPDEs. Our focus is on the convergence of the form $\int_0^T V_n\, \mathrm{d} W_n \to \int_0^T V\, \mathrm{d} W$, where $\{V_n\}$ is bounded in $L^p(Ω\times [0,T];X)