stochastic processes

Consider a linear growth process with immigration rate k where λn = nλ + k and µn = nµ

i. Write down the difference - differential equation for this process. 

i. Show that M(t) = E{X(t)} for this process satisfies the differential equation M0(t) = (λ−µ)M(t) + k i. 

Hence show that with initial condition M(0) = i if X(0) = i, Then M(t) = k λ−µ{e(λ−µ)t −1}+ ie(λ−µ)t, λ 6= µ