09-FEB-2025
This paper makes four contributions to the Langevin-Zamrik (LZ) interest rate model. **(1) HJM identification:** we show that LZ is a Gaussian Heath-Jarrow-Morton model with closed-form volatility $\sigma^{\mathrm{HJM}}(\tau) = -(\sigma/m)B_2′(\tau)$, where no-arbitrage is automatic rather than imposed. **(2) Three-regime volatility:** the overdamped, critically damped, and underdamped regimes produce humped, Nelson-Siegel, and oscillatory volatility shapes respectively; the underdamped volatility changes sign, a structural feature impossible in any existing Gaussian HJM family. The no-arbitrage drift is fully closed-form in all three regimes. **(3) Inertial Musiela SPDE:** LZ is lifted to an infinite-dimensional second-order SPDE on the space of forward curves; it collapses back to the 2D LZ system via the Björk-Christensen finite-dimensional realisation theorem, with the critically damped case yielding the no-arbitrage Nelson-Siegel family. **(4) Forward Rate LZ PDE:** the central new result. A change of coordinates from $(r_t,v_t)$ to any two market-observable forward rates $(F_1, F_2)$ transforms the LZ PDE into one whose state variables are directly quoted in swap markets and whose coefficients are exactly the HJM drift and volatility. The original LZ PDE is the special case $x_1=0$, $x_2\to 0$.