Exercise . Suppose x = (xt) is an affine diffusion with dynamics dxt = ( xt)

Exercise . Suppose x = (xt) is an affine diffusion with dynamics dxt = (ϕ − κxt) dt + !

δ + δxt dzt.

Then Section . has shown that Et

, e

−

T t xu du-

= e

−a(T−t)−b(T−t)xt for functions a and b that solve a system of ordinary differential equations. Show that Et

, e

−

T t xu du(ν + νxT)

-

= (aˆ(T − t) + ˆ

b(T − t)xt)e

−a(T−t)−b(T−t)xt , where a and b are the same as above, and where aˆ and ˆ

b are also deterministic functions. Provide a system of ordinary differential equations (with appropriate boundary conditions) which aˆ and ˆ

b must solve. Hint: First note that Et

, exp{−

T t xu du}(ν + νxT)

-

can be written as f(xt, t) for some function f .

Use Theorem . to write down a partial differential equation for f and verify that f(x, t) = (aˆ(T − t) + ˆ

b(T − t)x) exp{−a(T − t) − b(T − t)x} is a solution when the deterministic functions solve appropriate ordinary differential equations.