why do we say Vasicek interest rate model has a closed form solution

Question

Vasicek interest rate model solution has the form of:

and Cox-Ingersoll-Ross (CIR) interest rate model solution has the form of:

As we can see both models have dW term at the end, why do we say that Vasicek interest rate model has a closed form solution ?

Answer 1

The Vasicek solution is closed form because the stochastic integral

$$I(a)= \int_0^t e^{as} \, dW_s$$

has a deterministic integrand and the moments can be obtained in closed form

$$E[I(a)] = 0, \\ \text{var}[I(a)] = \int_0^t e^{2as} \, dt = \frac{e^{2at}-1}{2a}$$