For an Ito-Levy process $X_t$ with the dynamics given by $$ \mathrm{d}X_t = \sigma_t \mathrm{d}B_t + \int_{|z|<1}\gamma_t(z) \tilde{N}(\mathrm{d}t, \mathrm{d}z) + \int_{|z|\ge 1}\gamma_t(z) {N}(\mathrm{d}t, \mathrm{d}z), $$ where $B_t$ is a standard Brownian motion, $\tilde{N}(\mathrm{d}t, \mathrm{d}z)$ is the compensated Poisson random measure and ${N}(\mathrm{d}t, \mathrm{d}z)$ is the Poisson random measure. I am trying to prove the following theorem (source: Th 1.17 of Applied Stochastic Control of Jump Diffusions by Agnès Sulem and Bernt Øksendal): $$ \mathbb{E}\left[X^2_T \right] = \mathbb{E}\left[ \int_0^T \left\{\sigma^2_t + \int_{\mathbb{R}} \gamma^2_t(z) \nu(\mathrm{d}z) \right\} \mathrm{d}t\right]. $$
Here's what I did: Applying Ito's formula to the process $X_t^2$, we obtain $$ \begin{align*} \mathrm{d}X_t^2 &= 2X_t\sigma_t \mathrm{d}B_t + \sigma_t^2 \mathrm{d}t + \int_{|z|<1}\left\{(X_{t^-}+\gamma_t(z))^2 -X_{t^-}^2 - 2X_{t^-}\gamma_t(z) \right\}\nu(\mathrm{d}z)\mathrm{d}t\\ &~~~+\int_{z|<1}\left\{(X_{t^-}+\gamma_t(z))^2 -X_{t^-}^2\right\} \tilde{N}(\mathrm{d}t, \mathrm{d}z)+ \int_{z|\ge 1}\left\{(X_{t^-}+\gamma_t(z))^2 -X_{t^-}^2\right\} {N}(\mathrm{d}t, \mathrm{d}z)\\ &=2X_t\sigma_t \mathrm{d}B_t + \sigma_t^2 \mathrm{d}t +\int_{z|<1}\left\{(X_{t^-}+\gamma_t(z))^2 -X_{t^-}^2\right\} \tilde{N}(\mathrm{d}t, \mathrm{d}z)\\ &~~~+ \int_{|z|<1} \gamma_t^2(z) \nu(\mathrm{d}z)\mathrm{d}t +\int_{z|\ge 1}\left\{\gamma_t^2(z)+2X_{t^-}\gamma_t(z) \right\} {N}(\mathrm{d}t, \mathrm{d}z)\\ &=2X_t\sigma_t \mathrm{d}B_t + \sigma_t^2 \mathrm{d}t +\int_{z|<1}\left\{(X_{t^-}+\gamma_t(z))^2 -X_{t^-}^2\right\} \tilde{N}(\mathrm{d}t, \mathrm{d}z)\\ &~~~+ \int_{|z|<1} \gamma_t^2(z) \nu(\mathrm{d}z)\mathrm{d}t +\int_{z|\ge 1}\left\{\gamma_t^2(z)+2X_{t^-}\gamma_t(z) \right\} \left(\tilde{N}(\mathrm{d}t, \mathrm{d}z)+\nu(\mathrm{d}z)\mathrm{d}t \right). \end{align*} $$ Then by integration and taking expectation, the term contains $B_t$ and $\tilde{N}$ vanish. However, from my derivation, there is an additional term $\mathbb{E}[\int_0^T \int_{z|\ge 1}2X_{t^-}\gamma_t(z)\nu(\mathrm{d}z)\mathrm{d}t]$ in the final expression of $\mathbb{E}\left[X^2_T \right]$.
I am not sure where went wrong. Any help would be greatly appreciated.