The jump problem for the laplace equation

AbstractThe boundary value problem for the Laplace equation outside several cuts in a plane is studied. The jump of the solution of the Laplace equation and the jump of its normal derivative are specified of the cuts. The problem is studied under different conditions at infinity, which lead to different uniqueness and existence theorems. The solution of this problem is constructed in the explicit form by means of single-layer and angular potentials. The singularities at the ends of the cuts are investigated

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

Calculation of electrostatic fields using quasi-Green's functions: application to the hybrid Penning trap.

Penning traps offer unique possibilities for storing, manipulating and investigating charged particles with high sensitivity and accuracy. The widespread applications of Penning traps in physics and chemistry comprise e.g. mass spectrometry, laser spectroscopy, measurements of electronic and nuclear magnetic moments, chemical sample analysis and reaction studies. We have developed a method, based on the Green's function approach, which allows for the analytical calculation of the electrostatic properties of a Penning trap with arbitrary electrodes. The ansatz features an extension of Dirichlet's problem to nontrivial geometries and leads to an analytical solution of the Laplace equation. As an example we discuss the toroidal hybrid Penning trap designed for our planned measurements of the magnetic moment of the (anti)proton. As in the case of cylindrical Penning traps, it is possible to optimize the properties of the electric trapping fields, which is mandatory for high-precision experiments with single charged particles. Of particular interest are the anharmonicity compensation, orthogonality and optimum adjustment of frequency shifts by the continuous SternGerlach effect in a quantum jump spectrometer. The mathematical formalism developed goes beyond the mere design of novel Penning traps and has potential applications in other fields of physics and engineering

The first-crossing area of a diffusion process with jumps over a constant barrier

For a given barrier $S$ and a one-dimensional jump-diffusion process $X(t),$ starting from $x<S,$ we study the probability distribution of the integral $A_S(x)= \int_0 ^ {\tau_S(x)}X(t) \ dt$ determined by $X(t)$ till its first-crossing time $\tau_S(x)$ over $S.$ In particular, we show that the Laplace transform and the moments of $A_S(x)$ are solutions to certain partial differential-difference equations with outer conditions. The distribution of the minimum of $X(t)$ in $[0, \tau_S(x)]$ is also studied. Thus, we extend the results of a previous paper by the author, concerning the area swept out by $X(t)$ till its first-passage below zero. Some explicit examples are reported, regarding diffusions with and without jumps

Extreme times in financial markets

We apply the theory of continuous time random walks to study some aspects of the extreme value problem applied to financial time series. We focus our attention on extreme times, specifically the mean exit time and the mean first-passage time. We set the general equations for these extremes and evaluate the mean exit time for actual data.Comment: 6 pages, 3 figure

Universality classes in Burgers turbulence

We establish necessary and sufficient conditions for the shock statistics to approach self-similar form in Burgers turbulence with L\'{e}vy process initial data. The proof relies upon an elegant closure theorem of Bertoin and Carraro and Duchon that reduces the study of shock statistics to Smoluchowski's coagulation equation with additive kernel, and upon our previous characterization of the domains of attraction of self-similar solutions for this equation

Exit times in non-Markovian drifting continuous-time random walk processes

By appealing to renewal theory we determine the equations that the mean exit time of a continuous-time random walk with drift satisfies both when the present coincides with a jump instant or when it does not. Particular attention is paid to the corrections ensuing from the non-Markovian nature of the process. We show that when drift and jumps have the same sign the relevant integral equations can be solved in closed form. The case when holding times have the classical Erlang distribution is considered in detail.Comment: 9 pages, 3 color plots, two-column revtex 4; new Appendix and references adde

Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation

A detailed study is presented for a large class of uncoupled continuous-time random walks (CTRWs). The master equation is solved for the Mittag-Leffler survival probability. The properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed. Finally, some common objections found in the literature are thoroughly reviewed.Comment: Preprint version of an already published paper. 8 page

Unified Solution of the Expected Maximum of a Random Walk and the Discrete Flux to a Spherical Trap

Two random-walk related problems which have been studied independently in the past, the expected maximum of a random walker in one dimension and the flux to a spherical trap of particles undergoing discrete jumps in three dimensions, are shown to be closely related to each other and are studied using a unified approach as a solution to a Wiener-Hopf problem. For the flux problem, this work shows that a constant c = 0.29795219 which appeared in the context of the boundary extrapolation length, and was previously found only numerically, can be derived explicitly. The same constant enters in higher-order corrections to the expected-maximum asymptotics. As a byproduct, we also prove a new universal result in the context of the flux problem which is an analogue of the Sparre Andersen theorem proved in the context of the random walker's maximum.Comment: Two figs. Accepted for publication, Journal of Statistical Physic