This paper contains a survey of results about linear and nonlinear partial differential equations of Kolmogorov type arising in physics and in mathematical finance. Some recent pointwise estimates proved in collaboration with S. Polidoro are also presented. Mathematics Subject Classification (2000). AMS Subject Classification: 35K57, 35K65, 35K70.

The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two mai...

In this paper we study the existence of periodic solutions of the fourth-order equations u − pu′′ − a x u + b x u3 = 0 and u − pu′′ + a x u − b x u3 = 0, where p is a positive constant, and a x and b x are continuous positive 2Lperiodic functions. The boundary value problems P1 and P2 for these equations are considered respectively with the boundary conditions u 0 = u L = u′′ 0 = u′′ L = 0. Exi...

Solvability of forward–backward stochastic di erential equations with nonsmooth coe cients is considered using the Four-Step Scheme and some approximation arguments. For the onedimensional case, the existence of an adapted solution is established for the equation which allows the di usion in the forward equation to be degenerate. As a byproduct, we obtain the existence of a viscosity solution t...

A new, wide class of relativistic stochastic processes is introduced. All relativistic processes considered so far in the literature (the Relativistic Ornstein-Uhlenbeck Process as well as the Franchi-Le Jan and the Dunkel-Hänggi processes) are members of this class. The stochastic equations of motion and the associated forward Kolmogorov equations are obtained for each process in the class. Th...

This paper discusses the pricing of CDOs in a Markov chain framework. We show that in general the values of the legs satisfy systems of partial differential equations. In the special case of constant default intensities, one only needs to solve a system of ordinary differential equations, the so-called Kolmogorov differential equations.

We study the null controllability of Kolmogorov-type equations ∂t f + v ∂x f − ∂2 v f = u(t, x, v)1ω(x, v) in a rectangle , under an additive control supported in an open subset ω of . For γ = 1, with periodic-type boundary conditions, we prove that null controllability holds in any positive time, with any control support ω. This improves the previous result by Beauchard and Zuazua (Ann Ins H P...

We study a class of elliptic operators A with unbounded coefficients defined in I × R for some unbounded interval I ⊂ R. We prove that, for any s ∈ I, the Cauchy problem u(s, ·) = f ∈ Cb(R ) for the parabolic equation Dtu = Au admits a unique bounded classical solution u. This allows to associate an evolution family {G(t, s)} with A, in a natural way. We study the main properties of this evolut...

In a series of papers [1]–[4] we considered parabolic equations for measures on R. Our motivation was a study of the Kolmogorov equations for transition probabilities of diffusion processes. Here we are concerned with similar problems in infinite dimensions. Applications are given to stochastic partial differential equations such as stochastic equations of Navier– Stokes and reaction-diffusion ...