Feynman-Kac for dividend stream

Question

Suppose an asset value process $V_t$ that solves the PDE $$dV_t=\mu V_tdt+\sigma V_tdW_t \text{ with }\mu\in\mathbb{R},\sigma>0, W \text{ Brownian Motion}.$$ I want to price a dividend stream $D_T=\int _0 ^Td(V_s)ds$ for a bounded function $d(v)$. Therefore I need to derive the conditional expectation: $$E_{(t,V_t)}\left(\int_t ^T e^{-r(s-t)}d(V_s)ds\right)$$ for constant $r>0$. I fail in deriving the necessary PDE to ensure that the value process is a (local) martingale. As a solution I have given the following: Suppose that the bounded function $F:[0,T]\times \mathbb{R}^+\rightarrow\mathbb{R}$ satisfies the PDE $$F_t+\mu vF_v+\frac{1}{2}\sigma^2v^2F_{vv}+d(t,v)=rF(t,v)$$ with terminal condition $F(T,v)=0$. I try the following: Let $Z(t,V)=\int_t ^T e^{-r(s-t)}d(V_s)ds=\int_t ^T Y_s ds.$ Following Itô it can be shown that $dY_s=e^{-r(s-t)}[(-rd+d_t+d_v\mu v+\frac{1}{2}d_{vv}\sigma^2v^2)dt+d_vv\sigma dW_t]$. However, how can I write $dZ=Z_tdt+Z_YdY+\frac{1}{2}Z_{YY}d{Y}$?