= Forward-curve calibration of a shifted Brownian short rate
{title2=$g(T)=f_0(T)+\frac12\sigma^2T^2$}
To match a prescribed initial <instantaneous forward rate> curve $F_0(T)$ in the <Gaussian short-rate model with a deterministic shift>, choose
$$
g(T)=F_0(T)+\tfrac12\sigma^2T^2.
$$
Since $W_0^Q=0$, its model forward rate is then $f_0(T)=g(T)-\sigma^2T^2/2=F_0(T)$. Moreover,
$$
P_0(T)=\exp\left[-\int_0^Tg(s)ds+\frac{\sigma^2T^3}{6}\right]=\exp\left[-\int_0^TF_0(s)ds\right],
$$
so the initial <zero-coupon bond> curve matches as well. A continuous initial forward curve gives pointwise calibration; a locally integrable curve gives the corresponding almost-everywhere interpretation of the <instantaneous forward rate>.
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