Stochastic calculus

Problem 5. Interest rate model (30 pts) In this problem, we consider a Brownian motion (Wt)(20. We consider a model for interest rates (X )teo given as the solution to the SDE: dXt = --Xtdt + dWt, with Xo = 1. (a) We are interested in finding E[X} | Fi] := g(t, X:). Write a PDE with terminal condition that g must satisfy. (b) Determine the function g by solving the PDE. Hint: try a function of the form g(t, x) = a(t)x- + b(t)x + c(t). (c) Use the transformation Yr = eat X, with an appropriate choice of the parameter a to determine an explicit expression for the process (Xt). (d) From the result in (c), determine E[X} | F]. If you couldn't solve (c), use X = et (1+ edW. ) for the questions below. 2 (e) Determine lim Var(Xt). 1-+00 (f) We claim that the process (Xt) remains non-negative with probability one. True or False ? (Explain your answer.)