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The differential equation has a trend due to the interest rate. When you discount you take this trend away: $$\frac{d}{dt} (e^{-rt}Z_t) = -re^{-rt}Z_t + e^{-rt} \frac{d}{dt}Z_t = e^{-rt}\frac{1}{2}S_t^2\Gamma_t(\hat{\sigma}^2-\beta_t^2)$$ $Z$ doesn't appear on the rhs anymore and you can integrate  e^{-rT}Z_T - e^{-r0}Z_0 = \int_0^T ...