 Static terminal-payoff representation of a down-and-in claim

For initial spot $s>b$ in the Black-Scholes model, put $\nu =(\rho -{\sigma }^2/2)/\sigma$ and $\kappa =(s/b{)}^2$. A down-and-in claim with terminal payoff $f$ has the same initial price as the terminal claim $g(x)=f(x)1_{x\le b}+{\kappa }^{-\nu /\sigma }f(x/\kappa )1_{x>\kappa b}$. Reflection of the log-price endpoint above the lower barrier changes its normal probability density to $e^{2\nu \ell }{\varphi }_T(y-2\ell -\nu T)$, where $\ell =\log (b/s)/\sigma$. Substitution $z=y-2\ell$ gives the payoff identity. This is price equality, not pathwise equality of payments.