The list of main results is as follows

1. Consistency conditions for ATSMs. The standard procedure for pricing of contingent claims in affine term structure models (ATSMs) is based to reduction to a boundary problem for the corresponding parabolic equation, and solution of the boundary problem using the Fourier transform and solve a family of differential Riccati equations. The reduction is typically justified by the reference to the Feynman-Kac theorem. However, in ATSMs, the zero order term (``electric potential", that is, the interest rate) of the operator is linear, hence, in many models, non-semibounded. Therefore, the standard form of the Feynman-Kac theorem is not applicable. In the following two papers, the Feynman-Kac theorem in the case of non-semibounded ``electric potential" is reduced to the case of semi-bounded electric potential. In addition, the sets of conditions, which ensure the monotonicity of the bond price as a function of time, are derived.

• Sergei Levendorskiĭ, Consistency conditions for affine term structure models, Stochastic Processes and Their Applications 109 (2004), 225-261 pdf file
• Sergei Levendorskiĭ, Consistency conditions for Affine Term Structure Models II. Option pricing under diffusions with embedded   jumps, Annals of Finance (2006), 2:2, 207-224 http://ssrn.com/abstract=627622

2. Asymptotic pricing in QTSMs with jumps. ATSMs with jumps are as tractable as ATSMs without jumps, and in many cases, closed-from solutions are available. In quadratic term structure models, closed-form solutions are possible only if there are no jumps. In

•  Sergei Levendorskiĭ, Pseudo-diffusions and Quadratic Term Structure Models; Mathematical Finance   (2005), 15, 393-424. http://ssrn.com/abstract=520044

QTSMs with jumps (the case of no diffusion term being allowed) are considered, and asymptotic formulas are derived in the form of series w.r.t. the inverse of large parameter, which characterizes the rate of decay of distribution of jumps.

In the following paper, several asymptotic expansions are derived for the case of jumps of small intensity

• Nina Boyarchenko and Sergei Levendorskiĭ, Asymptotic Pricing in Term Structure Models Driven by Jump-Diffusions of Ornstein-Uhlenbeck Type (March 14, 2006)   http://ssrn.com/abstract=890725

One of the expansions is based on

3. Eigenfunction expansion in multi-factor models or diagonalization of non-self-adjoint quadratic Hamiltonians.  This procedure is used in

• Nina Boyarchenko and Sergei Levendorskiĭ, The eigenfunction expansion method in multi-factor quadratic term structure models,  Mathematical Finance 17, No. 4 (2007), pp.503–539 http://ssrn.com/abstract=874839

to derive efficient pricing formulas for contingent claims of European type. Significant advantages of this approach for contingent claims even of relatively short maturity is demonstrated.

4. Estimating equations for a class of time-irreversible multi-factor models. The diagonalization of the pricing semigroup constructed in the above paper is used in

• Nina Boyarchenko and Sergei Levendorskiĭ,Estimating equations for a class of time-irreversible multi-factor models, February, 2008,  http://ssrn.com/abstract=1088922  

to extend the operator approach to estimation of time-iireversible term structure models.

5. Problems for FFT-Riccati equations method in QTSM. In

• Nina Boyarchenko and Sergei Levendorskiĭ, On errors and bias of Fourier transform methods in quadratic term structure models,  International Journal of Theoretical and Applied Finance  10, No. 2 (2007) 273-306  http://ssrn.com/abstract=874835

it is demonstrated that a straightforward application of the standard approach to pricing in Gaussian QTSMs leads to systematic errors, and useful improvements are suggested.