 Exponentially weighted consumption habit

An exponentially weighted consumption habit is $h_t=e^{-\lambda t}{h}_0+{\int }_0^t\lambda e^{-\lambda (t-s)}{c}_{s}ds$. Differentiation gives $d{h}_t=\lambda (c_t-h_t)dt$. The state remains positive from positive initial habit and nonnegative consumption.