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

  • $\begingroup$ 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 $F_T$. $\endgroup$ – ilovevolatility Jun 13 '19 at 6:47
  • $\begingroup$ Does this answer your question? $\endgroup$ – Gordon Jun 14 '19 at 16:19

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