= Musiela forward-curve equation
{c}
{title2=$dg_t(x)=(\partial_xg_t(x)+A(x))dt+\langle\sigma(x),dW_t\rangle$}
For time-to-maturity coordinates $g_t(x)=f(t,t+x)$ and time-homogeneous deterministic <Heath-Jarrow-Morton model> volatility, $A(x)=\langle\sigma(x),\int_0^x\sigma(u)du\rangle$. The displayed evolution contains the maturity shift. Its mild solution is $g_t(x)=g_0(x+t)+\int_0^t A(x+s)ds+\int_0^t\langle\sigma(x+t-s),dW_s\rangle$. 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.
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