I am facing a problem in stochastic control and wonder if my method is right. I will try to simplify my situation. First I have this GBM:
$dY_t = \mu_Y Y_t dt+ \sigma_Y Y_t dB_1(t)$ And obviously $Y_t=Y_0 \exp\left[ \mu_Y t - \frac{\sigma_Y^2}{2} t +\sigma_Y B_1(t) \right]$
Using the Clark-Ocone Representation Theorem:
$Y_t=Y_0 \exp\left( \mu_Y t \right) + \sigma_Y \int_{0}^{t} Y_t dB(t)$
The state variable evloves like
$dX_t = rX_t dt + Y_t dt$
Here, I try to use Fubini's Theorem:
$\begin{aligned} dX_t &= rX_t dt + Y_0 \exp\left( \mu_Y t \right) dt + \sigma_Y \int_{0}^{t} Y_t dB(t) dt \\ & = rX_t dt + Y_0 \exp\left( \mu_Y t \right) dt + \sigma_Y \int_{0}^{t} Y_t dt dB(t) \end{aligned}$
In this way, I am able to proceed the task I face. So, I am wondering if the Fubini use is correct, please help me check it. Many thanks!