Other versions of my names: Sergey, Serge; Levendorskiy, Levendorski

My current research interests are pricing of financial options of various kind, mainly barrier options and American options on stocks, indices and exchange rates; real options; interest rate derivatives; estimation of time-irreversible processes. I am planning to work on credit risk models as well since this topic has become extremely important, and the standard approaches to credit risk proved to be inadequate. In addition, new regulation may drastically change the field, which will create numerous opportunities. See the related publication on Eurointelligence website http://www.eurointelligence.com/article.581+M5f964ea3344.0.html

and the working paper Boyarchenko, Svetlana I. and Levendorskii, Sergei Z.,Snowball Effect of a CDS Market(July 28, 2009). Available at SSRN: http://ssrn.com/abstract=1440388

 The complexity of financial markets requires sound understanding of the basic principles of economics and advanced mathematical tools from, essentially, all fields of mathematics. The former are needed to avoid or, at least, mitigate disasters such as the current credit crisis; analytical methods are indispensible for construction of efficient numerical methods. Interactions among methods originated in Stochastics (probabilistic version of the Wiener-Hopf factorization, Monte Carlo simulations) and the methods of Real and Complex Analysis (operator form of the Wiener-Hopf factoriztion, Integral  Transform  methods (Fourier transform  in particular), eigenfunction expansions, finite difference methods, splines, residues) are especially promising. To a great extent, my current research builds upon my past experience in Economics, Analysis and Algebra. For details, see the links on the left.

Past research: Economics.  Several years ago, my co-authors and I constructed two models of Russia's virtual economy. The first one, constructed before the August 1998 default, explained why Russia's virtual economy could not be reformed by monetary policies, and why the stabilization could not be achieved. The second model constructed after the default, explained why the prices in roubles had not changed much after the default but the prices in money substitutes had doubled. I have received the Fulbright award and Zvi Griliches award for the best research paper of the year for these models. The results were presented at a plenary talk at the international conference on Transition Economies (Moscow, 2000) and at a Money, Macro, and Finance meeting in London, 2000

Past research: Analysis. I worked in several fields of analysis: general theory of pseudo-differential  operators (PDO) including degenerating elliptic operators and hypoelliptic operators, spectral asymptotics for these operators with applications to Schrödinger operators. Among the results, a general method for calculation of spectral asymptotics for wide classes of PDE and PDO, generalizations of the classical Weyl formula and Colin de Verdiere's formula based on the method of orbits, spectral asymptotics for perturbed periodic Schrödinger operator, the proof of absolute continuity of the spectrum of wide classes of operators with periodic potentials (with Peter Kuchment), a general calculus of pseudo-differential operators with the symbols having singularities or degenerating at the boundary, and a calculus of boundary problems for degenerate elliptic operators.

Past research: Quantum groups. The main results, which I obtained, are: last step in classification of irreducible representations of compact quantum groups; proof of the multiplicative formula for the universal R-matrix for quantized universal enveloping algebras of  simple Lie groups  (with Yan Soibel’man); proof of the Poincare-Birkhoff-Witt theorem for Yangians; construction of Affine Yangians; a short proof of an important commutation relation for quantized enveloping algebras.