Term structure equation in the Vasicek model

Consider the SDE $$dr_t = (b-ar_t)dt +\sigma dW_t, \text{with } a; b > 0.$$ Let $$F(t; r) = E(\exp(-\int_{t}^{T}r_sds)| r_t = r).$$ (F can be interpreted as price of a zero coupon bond with maturity T.)

Use the Feynman-Kac formula to derive a PDE for the function $$F(t; r)$$.

I wanted to use Ito to obtain formula for $$r_t$$ and then plug it into Feynman-Kac formula $$E_{t0,x}=(\exp(-\int_{t}^{T}r(s,X_s)ds)\phi(X_t))$$ from the lecture, but I can't derive it. Any help is greatly appreciated

• Use Ito's formula to write down an expression for $dF_u$. Then note that $F_T - F_t = \int_t^T dF_u du$. Plug in the expression for $dF_u$ and note that $F_t$ is the expectation of $F_T$. – ilovevolatility Jun 13 at 6:47
• Does this answer your question? – Gordon Jun 14 at 16:19