Let $X_t$ and $Y_t$ be two Ito processes: $ dX_t = a_t dt + u_t dB_t$ and $ dY_t = c_t dt + v_t dB_t$. Then Ito's formula shows that $$ d(X_t Y_t) = X_t dY_t + Y_t dX_t + dX_t \cdot dY_t,$$ where the product $dX_t \cdot dY_t$ is computed according to the rule $$ dt \cdot dt =0,\quad dt\cdot dB_t = 0,\quad dB_t \cdot dB_t = dt. $$
Let $\alpha(t)$ be a function of bounded variation defined by $\alpha(t) = 1$ if $t = t_i$; $\alpha(t) = 0$ for $t \in [t_{i-1}, t_i)$; Let $N_t$ be a Poisson process and consider now the processes $$ dZ_t = a_t dt + d \alpha(t) + u_t dB_t + dN_t \quad \text{and}\quad dQ_t = c_t dt + d\alpha(t) + v_t dB_t + dN_t$$
My question is how does one calculate the product $d Y_t \cdot d Q_t$. This will involve stuff like $$ dt \cdot dN_t,\quad dt\cdot d \alpha, \quad d\alpha \cdot d\alpha,\quad d\alpha \cdot dN_t,\quad d\alpha \cdot dw,\quad dw\cdot dN_t, \quad dN_t \cdot dN_t $$