Monotonicity of the collateralized debt obligations term structure model

The problem of existence of arbitrage free and monotone CDO term structure models is studied. Conditions for positivity and monotonicity of the corresponding Heath-Jarrow-Morton-Musiela equation for the $x$-forward rates with the use of the Milian type result are formulated. Two state spaces are taken into account - of square integrable functions and a Sobolev space. For the first the regularity results concerning pointwise monotonicity are proven. Arbitrage free and monotone models are characterized in terms of the volatility of the model and characteristics of the driving L\'evy process.