I have an equation
$$S_{t} = B_{t}M_{t}-N_{t}$$
where $M_{t}$, $N_{t}$ are Geometric Brownian motions given by
$$dM_{t}=uM_{t}dt+\beta M_{t}dW_{t}$$ $$dN_{t}=nN_{t}dt+\alpha N_{t}dW_{t}$$ where $\mu$, $\beta$, $n$, and $\alpha$ are constants.
and $B_{t}$ is a function
$$B_{t} =\int_{t}^{T}\frac{1}{M_{s}}e^{rs}ds$$,
I would like to compute $dS_{t}$, If I compute it in the following way would it be correct?
$$dS_{t}=\frac{\partial B_{t}}{\partial t}M_{t}dt+B_{t}dM_{t}-dN_{t}$$