We study the interior regularity properties of the solutions of a nonlinear degenerate equation arising in mathematical finance. We set the problem in the framework of Hrmander type operators without assuming any hypothesis on the degeneracy of the associated Lie algebra. We prove that the viscosity solutions are indeed classical solutions. 2001 ditions scientifiques et mdicales Elsevier SAS Keywords: Nonlinear degenerate Kolmogorov equation, Interior regularity, Hrmander operators RSUM. -- Nous tudions la rgularit intrieure des solutions de viscosit d'une quation non linaire du second ordre dgnre que l'on rencontre en finance mathmatique. Nous tudions le problme par la thorie des oprateurs de Hrmander sans aucune hypothse sur la dgnerescence de l'algbre de Lie engendre. Nous montrons que la solution de viscosit est une solution classique. 2001 ditions scientifiques et mdicales Elsevier SAS 1.

This paper contains a survey on a series of papers by the authors, dealing with linear and non linear Kolmogorov-type operators, arising in diffusion theory, probability and finance. Some new results, about existence for Cauchy problems, regularity properties and pointwise estimates of solutions, are also announced.

We consider the following nonlinear degenerate parabolic equation which arises in some recent problems of mathematical finance:...

We prove some Schauder type estimates and an invariant Harnack inequality for a class of degenerate evolution operators of Kolmogorov type. We also prove a Gaussian lower bound for the fundamental solution of the operator and a uniqueness result for the Cauchy problem. The proof of the lower bound is obtained by solving a suitable optimal control problem and using the invariant Harnack inequality.

We prove Gaussian estimates from above of the fundamental solutions to a class of ultraparabolic equations. These estimates are independent of the modulus of continuity of the coefficients and generalize the classical upper bounds by Aronson for uniformly parabolic equations.

We adapt the iterative scheme by Moser...

Abstract. We characterize the domain of the realizations of the linear parabolic operator G defined by (1.4) in L 2 spaces with respect to a suitable measure, that is invariant for the associated evolution semigroup. As a byproduct, we obtain optimal L 2 regularity results for evolution equations with time-depending Ornstein-Uhlenbeck operators. 1.

We consider a class of ultraparabolic differential equations that satisfy the Hörmander’s hypoellipticity condition and we prove that the weak solutions to the equation with measurable coefficients are locally bounded functions. The method extends the Moser’s iteration procedure and has previously been employed in the case of operators verifying a further homogeneity assumption. Here we remove that assumption by proving some potential estimates and some ad hoc Sobolev type inequalities for solutions.

We give sufficient conditions for the discreteness of the spectrum of differential operators of the form Au = \Gamma\Deltau +hrF;rui in L 2 (R n ) where d(x) = e \GammaF (x) dx and for Schrodinger operators in L 2 (R n ). Our conditions are also necessary in the case of polynomial coefficients. Mathematics subject classification (1991): 35P05, 35J10, 35J70 1 Introduction In this paper we study the discreteness of the spectrum of two strictly related second order elliptic differential operators with unbounded coefficients on R n . These operators are A = \Gamma\Delta + n X i=1 @F @x i @ @x i ; B = \Gamma\Delta + V; with F 2 C 2 (R n ) and V 2 C(R n ). B is the classical Schrodinger operator, whereas A is a special case of second order operators with (possibly) unbounded coefficients of the first order terms. These operators are of interest when dealing with diffusion processes on all of R n in presence of a drift represented by the first order terms. Unlike...

The null controllability problem for a structurally damped abstract wave equation–a socalled elastic model–is considered with a view towards obtain optimal rates of blowup for the associated minimal energy function Emin(T), as terminal time T ↓ 0. Key use is made of the underlying analyticity of the elastic generator A, as well as of the explicit characterization of its domain of definition. We ultimately find that the blowup rate for Emin(T), as T goes to zero, depends on the extent of structural damping. 1