I've encountered the following equation:
$p_t = \pi_t + c \int_{-\infty}^te^{-\gamma(t-\tau)}dp_\tau, \quad 0<\gamma, 0<c<1$.
It is claimed that it can be rewritten as follows:
$p_t = \pi_t + \frac{c}{1-c}(\pi_t-\bar\pi_t)$,
where,
$\bar\pi_t = \lambda\int_{-\infty}^te^{-\lambda(t-\tau)}\pi_\tau d\tau, \quad \lambda = \frac{\gamma}{1-c}$
Do you have an idea why this is the case?
Any help is appreciated,