Petar Popivanov

Critical Gevrey Index for Hypoellipticity of Parabolic Operators and Newton Polygones (

SummaryIn this paper we give geometrical expressions of the (non) hypoellipticity in Gevrey spaces of parabolic operators via Newton polygones. We also determine the critical Gevrey class for which the hypoellipticity holds.

Gevrey Local Solvability for Degenerate Parabolic Operators of Higher Order

In this paper we study the local solvability in Gevrey classes for degenerate parabolic operators of order ≥ 2. We assume that the lower order term vanishes at a suitably smaller rate with respect to the principal part; we then analyze its influence on the behavior of the operator, proving local solvability in Gevrey spaces G s for small s, and local nonsolvability in G s for large s.

Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach

This work deals with the Dirichlet problem for some PDEs of second order with non-negative characteristic form. One main motivation is to study some boundary-value problems for PDEs of Black-Scholes type arising in the pricing problem for financial options of barrier type. Barrier options on stocks have been traded since the end of the Sixties and the market for these options has been dramatically expanding, making barrier options the most popular ones among the exotic. The class of standard barrier options includes ’in’ barriers and ’out’ barriers, which are activated (knocked in) and, respectively, extinguished (knocked out) if the underlying asset price crosses the barrier before expiration. Moreover, each class includes ’down’ or ’up’ options, depending on whether the barrier is below or above the current asset price and thus can be breached from above or below. Therefore there are eight types of standard barrier options, depending on their ’in’ or ’out’, ’down’ or ’up’, and ’call’ or ’put’ attributes. It is possible to include a cash rebate, which is paid out at option expiration if an ’in’ (’out’) option has not been knocked in (has been knocked out, respectively) during its lifetime. One can consider barrier options with rebates of several types, terminal payoffs of different forms (e.g. power options), more than one underlying assets and/or barriers, and allow for time-dependent barriers, thus enriching this class still further. On the other hand, a large variety of new exotic barriers have been designed to accommodate investors’ preferences. Another motivation for the study of such options is related to credit risk theory. Several credit-riskmodels build on the barrier option formalism, since the default event can be modeled throughout a signalling variable hitting a pre-specified boundary value (See [3],[8] among others). As a consequence, a substantial body of academic literature provides pricing methods for valuating barrier options, starting from the seminal work of [18], where an exact formula is offered for a down-and-out European call with zero rebate. Further extensions

Travelling Waves for Some Generalized Boussinesq Type Equations

This paper deals with traveling wave solutions of the (N+1)‐dimensional Boussinesq type equation and of the B(m,n) equation. They are found into explicit form being expressed in some special cases by the Jacobi elliptic function. In the general case they are written into integral form and it turns out that they develop cusp type singularities.

On the solvability of the characteristic Dirichlet problem for linear degenerate parabolic equations

We investigate the classical solvability for some classes of linear, degenerate equations in divergence form with prescribed Dirichlet data. Since the boundary value problem is characteristic according to Fichera on a part of the boundary, some typical nonlinear phenomena at these points are observed as boundary gradient blowups of the classical solutions in space directions. The regularity results explain the lack of hypoellipticity for special right-hand sides or boundary data for linear degenerate parabolic equations.