2. (30 Pts.) Vasicek Interest Rate Model The Vasicek interest rate model can be expressed in Langevin form dr(t) ab - r(t)) + f(t) dt where a represents speed of the reversion towards a mean, b is the long-term level of the mean, and o describes volatility. Assume &(t) is Gaussian white noise of zero mean. (a) Find an analytical solution as follows. Step 1: Apply the change of variables, r(t) = r(t) - b, noting that dx dr, and rewrite a new stochastic equation in x. Step 2: Let y = eatr and rewrite the SDE in x in terms of y. Step 3: Solve the resulting equation for y, then solve for x by reversing the change of variables, i.e., let r(t) = x(t) + b. (b) Solve numerically for r(t). (c) Compare analytical and numerical solutions. =