We investigate large changes, bursts, of the continuous stochastic signals, when the exponent of multiplicativity is higher than one. Earlier we have proposed a general nonlinear stochastic model which can be transformed into Bessel process with known first hitting (first passage) time statistics. Using these results we derive PDF of burst duration for the proposed model. We confirm analytical ...

We propose two main applications of Gyöngy (1986)’s construction of inhomogeneous Markovian stochastic differential equations that mimick the one-dimensional marginals of continuous Itô processes. Firstly, we prove Dupire (1994) and Derman and Kani (1994)’s result. We then present Bessel-based stochastic volatility models in which this relation is used to compute analytical formulas for the loc...

We consider a mixed problem with Dirichlet and integral conditions for a second-order hyperbolic equation with the Bessel operator. The existence, uniqueness, and continuous dependence of a strongly generalized solution are proved. The proof is based on an a priori estimate established in weighted Sobolev spaces and on the density of the range of the operator corresponding to the abstract formu...

We reexamine and continue the work of J. Vosmanskyý [23] on the concept of continuous ranking of zeros of certain special functions from the point of view of the transformation theory of second order linear differential equations. This leads to results on higher monotonicity of such zeros with respect to the rank and to the evaluation of some definite integrals. The applications are to Airy, Be...

Many important continuous-time Markov processes — for instance, the Ornstein-Uhlenbeck process and the Bessel processes — can be defined as solutions to stochastic differential equations with drift and diffusion coefficients that depend only on the current value of the process. The general form of such an equation (for a one-dimensional process with a one-dimensional driving Brownian motion) is...

In the present paper we prove that every k-dimensional Cartesian product of Kingman convolutions can be embedded into a k-dimensional symmetric convolution (k=1, 2, . . . ) and obtain an analogue of the Cramér-Lévy theorem for multi-dimensional Rayleigh distributions. A new class of multidimensional Rayleigh distributions and associated higher-dimensional Bessel processes are introduced and stu...

We show that the use of operational methods and of multi-index Bessel functions allow the derivation of generating functions, involving the product of an arbitrary number of Laguerre polynomials. ∗2000 Mathematics Subject Classification. 33C45, 44A45. †E-mail:[email protected] ‡E-mail:[email protected] §E-mail:[email protected] ¶E-mail:[email protected] 270 G. Dattol...

We propose a method to dynamically control the Fano resonance of a ring/crescent-ring gold nanostructure by spatially changing the phase distribution of a probe Bessel beam. We demonstrate that a highly tunable Fano interference between the quadrupole and bright dipole modes can be realized in the near-infrared range. Even though a complex interference between a broad resonance and a narrower r...

Image quality in light-sheet fluorescence microscopy is strongly affected by the shape of the illuminating laser beam inside embryos, plants or tissue. While the phase of Gaussian or Bessel beams propagating through thousands of cells can be partly controlled holographically, the propagation of fluorescence light to the detector is difficult to control. With each scatter process a fluorescence ...