Linear combinations of the telegraph random processes driven by partial differential equations
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KOLESNIK, Alexander. Linear combinations of the telegraph random processes driven by partial differential equations. In: Stochastics and Dynamics, 2018, nr. 4(18), p. 0. ISSN 0219-4937. DOI: https://doi.org/10.1142/S021949371850020X
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Stochastics and Dynamics
Numărul 4(18) / 2018 / ISSN 0219-4937

Linear combinations of the telegraph random processes driven by partial differential equations

DOI: https://doi.org/10.1142/S021949371850020X

Pag. 0-0

Kolesnik Alexander
 
Institute of Mathematics and Computer Science ASM
 
Disponibil în IBN: 7 decembrie 2018


Rezumat

Consider n independent Goldstein-Kac telegraph processes X1(t),...,Xn(t),n ≥ 2,t ≥ 0, on the real line ℝ. Each process Xk(t), k = 1,...,n, describes a stochastic motion at constant finite speed ck > 0 of a particle that, at the initial time instant t = 0, starts from some initial point xk0 = X k(0) ℝ and whose evolution is controlled by a homogeneous Poisson process Nk(t) of rate λk > 0. The governing Poisson processes Nk(t), k = 1,...,n, are supposed to be independent as well. Consider the linear combination of the processes X1(t),...,Xn(t), n ≥ 2, defined by L(t) =Σk=1na kXk(t), where ak, k = 1,...,n, are arbitrary real nonzero constant coefficients. We obtain a hyperbolic system of 2n first-order partial differential equations for the joint probability densities of the process L(t) and of the directions of motions at arbitrary time t > 0. From this system we derive a partial differential equation of order 2n for the transition density of L(t) in the form of a determinant of a block matrix whose elements are the differential operators with constant coefficients. Initial-value problems for the transition densities of the sum and difference S±(t) = X 1(t) ± X2(t) of two independent telegraph processes with arbitrary parameters, are also posed.

Cuvinte-cheie
determinant of block matrix, hyperbolic partial differential equations, initial-value problem, sum and difference of telegraph processes,

linear combinations, telegraph process, transition density