I am looking for general advice on how to start tackling the problem below. My background in math is fairly bad when it comes to stochastic differential equations, but if you have any recommendations that will guide me to the solution please let me know.

Consider the following process for prices: \begin{align} \frac{dX_t}{X_{t-}}&=\int_{\mathbb{R}^2}(e^x-1)\tilde{\mu}(dt, dx, dy)\\ dU_t&=-\alpha U_tdt+\mu\int_{\mathbb{R}^2}[(1-\rho)x^2\mathbb{1}_{\{x<0\}}+\rho y^2]\mu(dt, dx, dy)\\ \tilde{\mu}(dt, dx, dy)&=\mu(dt, dx, dy)-dt\otimes v_t(dx,dy)\\ \frac{v_t(dx,dy)}{dx dy}&=\begin{cases} c_te^{\lambda x}\mathbb{1}{\{x<0\}}+e^{-\lambda x}\mathbb{1}{\{x>0\}}, &\text{if }y=0\\ c_te^{\lambda y},&\text{if }x=0,y<0 \end{cases}\\ c_t&=U_{t-} \end{align} where $\mu$ is an integer-valued measure counting the jumps in the price $X$ and the state variable $U$, the corresponding jump intensity is $dt\otimes v_t(dx,dy)$, and $\tilde{\mu}(dt, dx, dy)=\mu(dt,dx,dy)-dtv_t(dx,dy)$ is the associated martingale jump measure.

Question: How should I approach computing $\mathbb{E}_t\left[\left(\frac{dX_t}{X_{t-}}\right)^2\right]$?

Thank you for your help! :D


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