  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.
