Default probability distributions are often defined in terms of their conditional default probability distribution, or their hazard rate. By their definition, they imply a unique probability density function. The applications of default probability distributions are varied, including the risk premium model used to price default bonds, reliability measurement models, insurance, etc. Fractional p...

An application of fractional Brownian motion (fBm) is considered in stochastic financial engineering models. For the known Fokker–Planck equation for fBm case, a solution transition probability density path integral method was built. It shown that mentioned does not result from Gaussian unit with precise covariance. expression approximation covariance found which solutions are based on measure ...

The class of fractional linear generating functions, one of the few known classes of probability generating functions whose iterates can be explicitly stated, is examined. The method of bounding a probability generating function g (satisfying g"(1)< oo) by two fractional linear generating functions is used to derive bounds for the extinction time distribution of the Galton-Watson branching proc...

Some properties of the fractional Schrödinger equation are studied. We prove the Hermiticity of the fractional Hamilton operator and establish the parity conservation law for fractional quantum mechanics. As physical applications of the fractional Schrödinger equation we find the energy spectra of a hydrogenlike atom (fractional "Bohr atom") and of a fractional oscillator in the semiclassical a...

Ideas from probability can be very useful to understand and motivate fractional calculus models for anomalous diffusion. Fractional derivatives in space are related to long particle jumps. Fractional time derivatives code particle sticking and trapping. This probabilistic point of view also leads to some interesting extensions, including vector fractional derivatives, and tempered fractional de...

Fractional calculus allows one to generalize the linear one-dimensional diiusion equation by replacing either the rst time derivative or the second space derivative by a derivative of fractional order. The fundamental solutions of these generalized diiusion equations are shown to provide probability density functions evolving in time or varying in space which are related to the special class of...

We extend the fractional genetic programming scheme with data elements that are no more scalar, but instead are similar to probability density functions. The extension straightforwardly fits into fractional programming, in which data elements are blended from several values. In the case of our previous work, the blend produced a single scalar value. The extension proposes to build an approximat...