Josep M. Porra

Some two and three-dimensional persistent random walks

Abstract We formulate non-Markovian versions of the persistent random walk in two and three dimensions and in continuous time. These models can be regarded as being generalizations of the original Pearson random walk and of the freely jointed chain which has been of some importance in polymer physics. Solutions for the probability density of the displacement of the random walker can be furnished for a restricted (essentially Markovian) set of these models. It is shown that in two dimensions the solution to one of these models is equivalent to the solution to an inhomogeneous telegraphers equation. It does not appear to be possible, starting from a similar model in three dumensions, to find any form of the telegraphers equation that follows from our solution of the equations in the Fourier-Laplace transform domains.

Properties of resonant activation phenomena

The phenomenon of resonant activation of a Brownian particle over a fluctuating barrier is revisited. We discuss the important distinctions between barriers that can fluctuate among ‘‘up’’ and ‘‘down’’ configurations, and barriers that are always ‘‘up’’ but that can fluctuate among different heights. A resonance as a function of the barrier fluctuation rate is found in both cases, but the nature and physical description of these resonances is quite distinct. The nature of the resonances, the physical basis for the resonant behavior, and the importance of boundary conditions are discussed in some detail. We obtain analytic expressions for the escape time over the barrier that explicitly capture the minima as a function of the barrier fluctuation rate, and show that our analytic results are in excellent agreement with numerical results. @S1063-651X~98!11304-1#

The continuous-time random walk description of photon motion in an isotropic medium

Lattice random walk models are useful alternatives to diffusion theory for describing photon migration in turbid media. Results obtained from random walk models were, in the past, based on approximating the propagator as a Gaussian function. This is equivalent to assuming that a large number of steps have been taken. We show that a particular, physically plausible, continuous-time random walk (CTRW) model can be solved exactly. At sufficiently long times the solution reduces to that for a random walk in discrete time. We provide exact solutions to problems related to cw reflection measurements and indicate some generalizations to time-gated measurements made on a uniform medium.

A dynamical model describing stock market price distributions

High-frequency data in finance have led to a deeper understanding on probability distributions of market prices. Several facts seem to be well established by empirical evidence. Specifically, probability distributions have the following properties: (i) They are not Gaussian and their center is well adjusted by Levy distributions. (ii) They are long-tailed but have finite moments of any order. (iii) They are self-similar on many time scales. Finally, (iv) at small time scales, price volatility follows a non-diffusive behavior. We extend Mertons ideas on speculative price formation and present a dynamical model resulting in a characteristic function that explains in a natural way all of the above features. The knowledge of such a distribution opens a new and useful way of quantifying financial risk. The results of the model agree – with high degree of accuracy – with empirical data taken from historical records of the Standard & Poors 500 cash index.

Black–Scholes option pricing within Itô and Stratonovich conventions

Options are financial instruments designed to protect investors from the stock market randomness. In 1973, Black, Scholes and Merton proposed a very popular option pricing method using stochastic differential equations within the Ito interpretation. Herein, we derive the Black–Scholes equation for the option price using the Stratonovich calculus along with a comprehensive review, aimed to physicists, of the classical option pricing method based on the Ito calculus. We show, as can be expected, that the Black–Scholes equation is independent of the interpretation chosen. We nonetheless point out the many subtleties underlying Black–Scholes option pricing method.

The continuum limit of a two-dimensional persistent random walk

Abstract We consider a two-dimensional persistent random walk in which the motion consists of alternative steps along one of two vectors, a and b . It is shown that the continuum limit of the evolution equation is not a two- but rather a one-dimensional telegraphers equation.

A diffusion model incorporating anisotropic properties

A mathematical model suggested by Orsingher, motivated by possible applicability to diffusion processes in which anisotropic scattering is significant, is reanalyzed. It is shown that the propagator for the full multi-dimensional model does not satisfy a telegraphers equation, as suggested by Orsingher, but that the propagator for the projection of motion on any one of the axes does. This can be used to analyze the results of a class of optical scattering measurements.

Demonstration of a conjecture for random walks in d-dimensional Sierpinski fractals

Random walks on some fractals can be analysed by renormalization procedures. These techniques make it possible to obtain the Laplace transform of the first-passage time probability density function of a random walker that moves in the fractal. The calculation depends on a function that is particular to each kind of fractal. For the Sierpinski family of fractals, it has been conjectured that , where d is the dimension of the Euclidean space in which the Sierpinski fractal is embedded. We provide a proof of the conjecture that is based on the symmetries of the Sierpinski fractal.

Isotropization time for non-Markovian CTRWs

We calculate the mean-squared displacement, , for a CTRW taking into account effects of anisotropic turn angles. When the pausing-time density is a negative exponential one finds a simple expression for which allows an exact determination of the transition time from ballistic to diffusive motion. In the non-Markovian case an exact expression is obtained for the Laplace transform of . The results are useful in the analysis of photon migration in a turbid medium.