I need some help with this question:

Let $B = (B_t)_{0\leq t \leq T}$ be a standard Brownian motion started at zero under a probability measure P, and let $\tilde{B} = (\tilde{B_t})_{0\leq t \leq T}$ be a stochastic process defined by

$$\tilde{B_t} = B_t + \int^{t}_0 B^2_s I(|B_s|\leq 1) ds $$

for $t\in [0,T]$ where $T>0$ is a given and fixed constant.

Compute $\tilde{E} ((\int^t_0\tilde{B_s} dB_s + \int^t_0 \tilde{B_s} B^2_s I(|B_s|\leq 1 ) ds)^2)$ for $t \geq 0$

Also, compute $\tilde{E}(e^{\sqrt{6}(B_\sigma + \int^\sigma_0 I(|B_s| \leq 1)ds)-3\sigma})$ and $\tilde{E}(e^{-3\sigma})$ when $\sigma = \inf\{t\geq 0 : B_t = 1 - \int^t_0 B^2_sI(|B_s|\leq 1)ds\}$

($\tilde{E}$ denotes expectation under $\tilde{P}$ and $E$ denotes expectation under $P$)


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