 Musiela forward-curve equation

For time-to-maturity coordinates $g_t(x)=f(t,t+x)$ and time-homogeneous deterministic Heath-Jarrow-Morton model volatility, $A(x)=⟨\sigma (x),{\int }_0^x\sigma (u)du⟩$. The displayed evolution contains the maturity shift. Its mild solution is $g_t(x)=g_0(x+t)+{\int }_0^{t}A(x+s)ds+{\int }_0^t⟨\sigma (x+t-s),d{W}_s⟩$. The whole curve is a time-homogeneous Markov process under suitable function-space well-posedness conditions; the current short rate alone need not be a sufficient state.