Q4. Bounded population growth is modeled as a stochastic differential equation dPrP(K- Pt)dt + aPtdWt, and Po is known. The constant K is the carrying capacity of the environment. The quantity rk is the per capita reproductive rate of the population at the low population density. The constant a is related to the size of the perturbation is population size due to random events in the environment. (a) Define Y = -1/P and show that the stochastic differential for Y is dY = [(a-rK)Y-r]dt - aYdWt. (b) Consider the stochastic differential equation dX = (a rK)Xdt aXdWt, - with initial condition Xo = 1. Show that the solution to X stochastic differential equa- tion is Xt=e(a/2-rK)t-aWi Q5. Solve the following PDE: OF OF 1 t -(t, x) + k (0 - x) 02F -(t, x) + 2 2 (t, x)- -TF = 0 F(T,x) = xekT where r, K, and are constants.