  &\hspace{0.0em}+ \int_t^{t_{k+0}}\int_{E} \beta_s^{\widetilde\tau,\zeta_{\alpha,\beta}(\omega)}(x) \left (\mu_{B^{\tau(\omega)}}(dx,ds) - F_s^{\widetilde{\tau},\zeta_{\alpha,\beta}(\omega)}(dx)ds\right) 