I try to compute the conditional expectation $E_{t}^{P}\left[B_{T}^{2} \ e^{-\frac{1}{2} \int_{0}^{T} \left( a + b B_{u}^{2}\right)^{2} d u+\int_{0}^{T} \left( a + b B_{u}^{2}\right) d B_{u}}\right]$ where
- $a$ and $b$ are deterministic
- $B_t$ is a P-Brownian motion
- $e^{-\frac{1}{2} \int_{0}^{T} \left( a + b B_{u}^{2}\right)^{2} d u+\int_{0}^{T} \left( a + b B_{u}^{2}\right) d B_{u}}$ is a P-martingale.
- $E_t$ means $E[...| \mathcal{F}_t]$
- There is equivalent Q measure $B^Q_t = B_t - \int_{0}^{t} \left( a + b B_{u}^{2}\right) du$
It seems difficult to work with Q-expectation as there will always a P-Brownian motion involved, e.g.
$ E^Q_t B_t = E^Q_t B^Q_t - E^Q_t \left(\int_{0}^{t} \left( a + b B_{u}^{2}\right) du \right)$
$ E^Q_t B^2_t = E^Q_t \left( B^Q_t - \int_{0}^{t} \left( a + b B_{u}^{2}\right) du \right)^2$
My thought is to use linear approximation $e^x = 1+x$
But I don't think my approach is correct. I was wondering if there is any clever way to derive this conditional expectation? I would really appreciate if anyone could show me how to solve this question.