A stochastic model related to the telegrapher's equation

Abstract

We will consider a very simple stochastic model, a random walk. Unfortunately, this model is little known. It has very interesting features and leads not to a diffusion equation but to a hyperbolic one. The model first appeared in the literature in a paper by Sidney Goldstein, known to you mostly because of his work in fluid dynamics. The model had first been proposed by G. I. Taylor -I think in an abortive, or at least not very successful, attempt to treat turbulent diffusion. But the model itself proved to be very interesting.The problem is the following: Suppose you have a lattice of points. I mean discrete, equidistant points as in Figure 1.

Telegraph Processes and Option Pricing

“…This naturally led to the telegraph equation describing the spatio-temporal dynamics of the potential in a transmitting cable, (without leakage) [13]. In his 1956 lecture notes, Kac, (see [14]), considered a continuous-time version of the telegraph model. Since then, the telegraph process and its various generalisations have been studied in great detail with numerous applications in physics, and, more recently, in financial market modelling.…”

Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics

“…However, this is at odds with many observations as well as with common sense [16]: since the body of animals of all species has the front end and the rear end, they are more likely to choose the new movement direction (following re-orientation) close to the movement direction at the preceding moment. The corresponding movement pattern is known as the correlated random walk (CRW) [17] and the corresponding microscopic stochastic process as the telegraph process [18], and their mean-field counterpart is known as a telegraph equation [19][20][21][22][23]:…”

“…The precise meaning of parameter τ can be slightly different depending on the details of the microscopic model; for instance, in the telegraph movement process, it is the time over which the animal moves without changing its movement direction [18][19][20]. Brownian motion thus corresponds to the limit τ → 0 when Equation (1) turns into the diffusion equation.…”

Supersymmetry in random two-velocity processes

“…The first direct treatment of the telegraph equation in continuous time as given in eq. (3) goes back to M. Kac [7] (see, [8] for a recent comprehensive overview on the history of random evolutions). As emphasised in [2,3], the persistent random walk provides a better description of spatial spread in population dynamics than Brownian motion.…”

“…(10) reduces to the standard telegraphist equation [6,7]. Since these pioneering works, numerous alternative derivations of Eq.…”