Probabilistic approach to solution of nonlinear PDES arising in financial mathematics

We apply a probabilistic approach developed in our previous papers, which allows us to solve the Cauchy problem for nonlinear parabolic equations and systems and to construct solutions of problems arising in financial mathematics in search for arbitrage-free option prices on nonideal markets. Bibliography: 11 titles.     

Characterization of elliptic distributions

We characterize elliptic distributions in ^d by an extremal property for a class of probability distances. A generalization to the case of Hilbert spaces is given. Bibliography: 2 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 19–28.Translated by Ya. I. Belopolskaya, L. B. Klebanov, and V. E. Volkovich.     

Calorimetric study of the native and postdenatured structures in starches with different degree of hydration

DSC studies of melting process of annealed native structures and postdenatured ones in low-amylose starches with different degrees of hydration were carried out. The starch recrystallization at different thermal treatments of the samples was studied both after the complete and partial destroy of native structures. It has been shown that native structures as well as postdenatured ones possess the ability to perfection, which is most clearly seen at the annealing at temperatures inside their melting ranges. The results obtained demonstrate that at the same duration of annealing the process of crystal perfection for secondary starch structures proceed more intensively compared to the native ones. The presence of the remained native structures in partial melt in contrast to the remained gel ones restricts the ability of the recrystallized structures to perfection.     

Generalized Solutions of Nonlinear Parabolic Systems and the Vanishing Viscosity Method

We show in this paper that stochastic processes associated with nonlinear parabolic equations and systems allow one to construct a probabilistic representation of a generalized solution to the Cauchy problem. We also show that in some cases the derived representation can be used to construct a solution to the Cauchy problem for a hyperbolic system via the vanishing viscosity method. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 7–39.     

Smooth Diffusion Measures and Their Transformations

In this paper, we show that some well-known equations of mathematical physics can be naturally treated from the probability viewpoint as equations for logarithmic derivatives of smooth diffusion measures. We discuss a probabilistic interpretation of the Darboux transformation applied in the theory of the SturmLiouville equations. It is shown that this transformation can be applied in deriving an explicit expression for the RadonNikodym derivative of a pair of absolutely continuous measures generated by various scalar multiplicative functionals of diffusion processes. Bibliography: 12 titles.     

Calorimetric Study of the Glass Transition Process in Humid Proteins and DNA

By method of differential scanning calorimetry the absolute values of heat capacity for the systemwater–biopolymer (globular and fibrillar proteins and DNA) were measured in a wide range of temperatures (from -30 up to 130°C) and concentrations of proteins both in native and denatured states. Thermal properties of humid denatured biopolymers demonstrate a characteristic anomaly in the form of the heat capacity jump at temperature depending on the bound water content. It has been shown that in the systems studied a glass transition, where water serves as a native plasticizer, is observed. It has been established that the S-shaped character of all heat capacity curves obtained on dehydration for native and denatured biopolymers is due to the gradual transition to the glassy state of both native and denatured samples. It was found that thermally denatured humid small globular proteins at subsequent dissolving in water at room temperature are able to restore their native structure. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stochastic processes in Q p associated with systems of nonlinear PIDEs

We construct a Markov process and its multiplicative operator functional associated with a system of nonlinear partial integral-differential equations (PIDE). As a result we derive a probabilistic representation of the solution to the Cauchy problem for a system of nonlinear PIDEs in Q _p . The text was submitted by the authors in English.