Characteristic Function of Time-Inhomogeneous L\'evy-Driven Ornstein-Uhlenbeck Processes

Distributional properties -including Laplace transforms- of integrals of Markov processes received a lot of attention in the literature. In this paper, we complete existing results in several ways. First, we provide the analytical solution to the most general form of Gaussian processes (with non-stationary increments) solving a stochastic differential equation. We further derive the characteristic function of integrals of L\'evy-processes and L\'evy driven Ornstein-Uhlenbeck processes with time-inhomogeneous coefficients based on the characteristic exponent of the corresponding stochastic integral. This yields a two-dimensional integral which can be solved explicitly in a lot of cases. This applies to integrals of compound Poisson processes, whose characteristic function can then be obtained in a much easier way than using joint conditioning on jump times. Closed form expressions are given for gamma-distributed jump sizes as an example.Comment: 15 pages, 26 pages, to appear in Statistics and Probability Letter

An Analysis of American Options under Heston Stochastic Volatility and Jump-Diffusion Dynamics

This paper considers the problem of pricing American options when the dynamics of the underlying are driven by both stochastic volatility following a square root process as used by Heston (1993), and by a Poisson jump process as introduced by Merton (1976). Probability arguments are invoked to find a representation of the solution in terms of expectations over the joint distribution of the underlying process. A combination of Fourier transform in the log stock price and Laplace transform in the volatility is then applied to find the transition probability density function of the underlying process. It turns out that the price is given by an integral dependent upon the early exercise surface, for which a corresponding integral equation is obtained. The solution generalises in an intuitive way the structure of the solution to the corresponding European option pricing problem in the case of a call option and constant interest rates obtained by Scott (1997).American options; stochastic volatility; jump-diffusion processes; Volterra integral equations; free boundary problem; method of lines

Laplace-Laplace analysis of the fractional Poisson process

We generate the fractional Poisson process by subordinating the standard Poisson process to the inverse stable subordinator. Our analysis is based on application of the Laplace transform with respect to both arguments of the evolving probability densities.Comment: 20 pages. Some text may overlap with our E-prints: arXiv:1305.3074, arXiv:1210.8414, arXiv:1104.404

Identification of Electric Charge Distribution Using Dual Reciprocity Boundary Element Models

Identification of unknown electric charges or sources distributed in space is made from the data observed over the field boundary using dual reciprocity boundary element models. The inhomogeneous term of the Poisson field can equivalently be expressed as the linear combination of the functions associated with the particular solutions to transform into Laplace equation. For the solution procedure, the variational formulation is employed, in which the regular boundary integral approach is incorporated to avoid the singularity. Numerical examples are presented to demonstrate the availability and the capability

Hilfer-Prabhakar Derivatives and Some Applications

We present a generalization of Hilfer derivatives in which Riemann--Liouville integrals are replaced by more general Prabhakar integrals. We analyze and discuss its properties. Further, we show some applications of these generalized Hilfer-Prabhakar derivatives in classical equations of mathematical physics, like the heat and the free electron laser equations, and in difference-differential equations governing the dynamics of generalized renewal stochastic processes