I'm trying to solve an equation for the stochastic exponent \begin{align} \mathcal{E}_t=\exp\left\{-\int_0^t b_s dB_s-\frac{1}{2}\int_0^t b_s^2dB_s\right\} \end{align} The process $(b_t)_{t\in\mathbb{R}_+}$ satisfies all conditions necessary for the exponential to be well defined.
Let \begin{align} Y_t:=-\int_0^t b_s dB_s \end{align} and its quadratic variation \begin{align} \langle Y\rangle_t=\int_0^tb_s^2ds \end{align}
For $p>1$ I'm trying to get
\begin{align} \mathcal{E}_t^{1-p}=\left(\exp\left\{pY_t-\frac{1}{2}\langle pY\rangle_t\right\}^{\frac{p-1}{p}}\right)\left(\exp\left\{\frac{(p+1)}{2}(p-1)p\langle Y\rangle_t\right\}^{\frac{1}{p}}\right) \end{align}
My problem is, that instead of obtaining the above equation, I calculate from the right-hand side and get \begin{align} \exp\left\{(p-1)Y_t-\frac{1}{2}(1-p)\langle Y\rangle_t\right\} \end{align} i.e. the sign on the $Y_t$ part is wrong.