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On the distribution of the maximum of the telegraph process
No full textIn this paper we present the distribution of the maximum of the telegraph process in the cases where the initial velocity is positive or negative with an even and an odd number of velocity reversals. For the telegraph process with positive initial velocity a reflection principle is proved to be valid while in the case of an initial leftward displacement the conditional distributions are perturbed by a positive probability of never visiting the half positive axis. Various relationships are established among the mentioned four classes of conditional distributions of the maximum. The unconditional distributions of the maximum of the telegraph process are obtained for positive and negative initial steps as well as their limiting behaviour. Furthermore the cumulative distributions and the general moments of the conditional maximum are presented
On the sojourn time of a generalized Brownian meander
No full textIn this paper we study the sojourn time on the positive half-line up to time t of a drifted Brownian motion with starting point u and subject to the condition that min0≤z≤lB(z)>v, with u>v. This process is a drifted Brownian meander up to time l and then evolves as a free Brownian motion. We also consider the sojourn time of a bridge-type process, where we add the additional condition to return to the initial level at the end of the time interval. We analyze the weak limit of the occupation functional as u↓v. We obtain explicit distributional results when the barrier is placed at the zero level, and also in the special case when the drift is null
On the exact distributions of the maximum of the asymmetric telegraph process
No full textIn this paper we present the distribution of the maximum of the asymmetric telegraph process in an arbitrary time interval [0,t] under the conditions that the initial velocity V(0) is either c1 or −c2 and the number of changes of direction is odd or even. For the case V(0)=−c2 the singular component of the distribution of the maximum displays an unexpected cyclic behavior and depends only on c1 and c2, but not on the current time t. We obtain also the unconditional distribution of the maximum for either V(0)=c1 or V(0)=−c2 and its expression has the form of series of Bessel functions. We also show that all the conditional distributions emerging in this analysis are governed by generalized Euler–Poisson–Darboux equations. We recover all the distributions of the maximum of the symmetric telegraph process as particular cases of the present paper. We underline that it rarely happens to obtain explicitly the distribution of the maximum of a process. For this reason the results on the range of oscillations of a natural process like the telegraph model make it useful for many applications
Some results on time-varying fractional partial differential equations and birth-death processes
Full text linkOn the fractional wave equation
No full textIn this paper we study the time-fractional wave equation of order 1 < n < 2 and give a probabilistic interpretation of its solution. In the case 0 < n < 1, d = 1, the solution can be interpreted as a time-changed Brownian motion, while for 1 < n < 2 it coincides with the density of a symmetric stable process of order 2/n. We give here an interpretation of the fractional wave equation for d > 1 in terms of laws of stable ddimensional processes. We give a hint at the case of a fractional wave equation for n > 2 and also at space-time fractional wave equations
Some Results on the Brownian Meander with Drift
No full textIn this paper we study the drifted Brownian meander that is a Brownian motion starting from u and subject to the condition that min ≤z≤t B(z) > v with u> v. The limiting process for u↓ v is analysed, and the sufficient conditions for its construction are given. We also study the distribution of the maximum of the meander with drift and the related first-passage times. The representation of the meander endowed with a drift is provided and extends the well-known result of the driftless case. The last part concerns the drifted excursion process the distribution of which coincides with the driftless case
The last zero-crossing of an iterated brownian motion with drift
No full textIn this paper, we consider the iterated Brownian motion μ1μ2I(t)=Bμ11(∣∣Bμ22(t)∣∣) where Bμjj,j=1,2 are two independent Brownian motions with drift μj. Here, we study the last zero crossing before the maximum time span travelled by the inner process of μ1μ2I(t) and for this purpose we derive the last zero-crossing distribution of the drifted Brownian motion. We derive also the joint distribution of the last zero crossing before t and of the first passage time through the zero level of a Brownian motion with drift μ after t. All these results permit us to derive explicit formulas for IμT0=sup{
Stochastic Dynamics of Generalized Planar Random Motions with Orthogonal Directions
No full textWe study planar random motions with finite velocities, of norm c > 0, along orthogonal directions and changing at the instants of occurrence of a nonhomogeneous Poisson process with rate function lambda = lambda(t), t >= 0. We focus on the distribution of the current position (X(t), Y(t)), t >= 0, in the case where the motion has orthogonal deviations and where also reflection is admitted. In all the cases, the process is located within the closed square S-ct = {(x, y) is an element of R-2 : |x| + |y| <= ct} and we obtain the probability law inside S-ct, on the edge & part;S-ct and on the other possible singularities, by studying the partial differential equations governing all the distributions examined. A fundamental result is that the vector process (X, Y) is probabilistically equivalent to a linear transformation of two (independent or dependent) one-dimensional symmetric telegraph processes with rate function proportional to lambda and velocity c/2. Finally, we extend the results to a wider class of orthogonal-type evolutions
Hitting Distribution of a Correlated Planar Brownian Motion in a Disk
Full text linkIn this article, we study the hitting probability of a circumference CR for a correlated Brownian motion B(t) = (B1 (t), B2 (t)), ρ being the correlation coefficient. The analysis starts by first mapping the circle CR into an ellipse E with semiaxes depending on ρ and transforming the differential operator governing the hitting distribution into the classical Laplace operator. By means of two different approaches (one obtained by applying elliptic coordinates) we obtain the desired distribution as a series of Poisson kernels
Pseudoprocesses related to higher-order equations of vibrations of rods
No full textIn this paper we study Fresnel pseudoprocesses whose signed measure density is a solution to a higher-order extension of the equation of vibrations of rods. We also investigate space-fractional extensions of the pseudoprocesses related to the Riesz operator. The measure density is represented in terms of generalized Airy functions which include the classical Airy function as a particular case. We prove that the Fresnel pseudoprocess time-changed with an independent stable subordinator produces genuine stochastic processes. In particular, if the exponent of the subordinator is chosen in a suitable way, the time-changed pseudoprocess is identical in distribution to a mixture of stable processes. The case of a mixture of Cauchy distributions is discussed and we show that the symmetric mixture can be either unimodal or bimodal, while the probability density function of an asymmetric mixture can possibly have an inflection point
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