Differential and integral equations for jump random motions
Authors:
A. O. Pogorui and R. M. Rodríguez-Dagnino
Journal:
Theor. Probability and Math. Statist. 101 (2020), 233-242
MSC (2020):
Primary 60K35; Secondary 60K99, 60K15
DOI:
https://doi.org/10.1090/tpms/1123
Published electronically:
January 5, 2021
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Additional Information
Abstract: In this paper we obtain a differential equation for the characteristic function of random jump motion on the line, where the direction alternations and random jumps occur according to the renewal epochs of the Erlang distribution. We also study random jump motion in higher dimensions and we obtain a renewal-type equation for the characteristic function of the process. In the 3-dimensional case we obtain the telegraph-type differential equation for jump random motion, where the direction alternations and random jumps occur according to the renewal epochs of the Erlang-2 distribution.
References
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References
- N. Ratanov, A jump telegraph model for option pricing, Quant. Finance 7 (2007), no. 5, 575–583. MR 2358921
- O. Lopez and N. Ratanov, Option pricing driven by a telegraph process with random jumps, J. Appl. Prob. 49 (2012), no. 3, 838–849. MR 3012103
- N. Ratanov, Option pricing model based on a Markov-modulated diffusion with jumps, Braz. J. Probab. Stat. 24 (2010), no. 2, 413–431. MR 2643573
- N. Ratanov and A. Melnikov, On financial markets based on telegraph processes, Stochastics 80 (2008), no. 2–3, 247–268. MR 2402167
- A. Di Crescenzo and B. Martinucci, On the generalized telegraph process with deterministic jumps, Methodology and Computing in Applied Probability 15 (2013), no. 1, 215–235. MR 3030219
- A. Di Crescenzo, A. Iuliano, B. Martinucci, and S. Zacks, Generalized telegraph process with random jumps, J. Appl. Probab. 50 (2013), no. 2, 450–463. MR 3102492
- A. Di Crescenzo, On random motions with velocities alternating at Erlang-distributed random times, Adv. Appl. Probab. 33 (2001), no. 3, 690–701. MR 1860096
- A. Pogorui and R. M. Rodríguez-Dagnino, One dimensional semi-Markov evolution with general Erlang sojourn times, Random Operators Stoch. Equ. 13 (2005), no. 4, 399–405. MR 2183564
- A. Pogorui and R. M. Rodríguez-Dagnino, Random motion with uniformly distributed directions and random velocity, J. Stat. Phys. 147 (2012), no. 6, 1216–1225. MR 2949527
- A. De Gregorio, On random flights with non-uniformly distributed directions, J. Stat. Phys. 147 (2012), no. 2, 382–411. MR 2922773
- A. De Gregorio, A family of random walks with generalized Dirichlet steps, J. Math. Phys. 55 (2014), no. 2, 023302. MR 3202881
- R. Garra and E. Orsingher, Random flights governed by Klein–Gordon–type partial differential equations, Stoch. Process. Appl. 124 (2014), no. 6, 2171–2187. MR 3188352
- A. Di Crescenzo and A. Meoli, On a jump-telegraph process driven by an alternating fractional Poisson process, J. Appl. Probab. 55 (2018), no. 1, 94–111. MR 3780385
- L. Angelani, Run-and-tumble particles, telegrapher’s equation and absorption problems with partially reflecting boundaries, J. Physics A: Mathematical and Theoretical 48 (2015), 495003. MR 3434824
- V. S. Korolyuk and A. V. Swishchuk, Semi-Markov Random Evolutions, Kluwer Academic Publishers, Dordrecht, 1995. MR 1472977
- V. S. Korolyuk and V. V. Korolyuk, Stochastic Models of Systems, Kluwer Academic Publishers, Dordrecht, 1999. MR 1753470
- A. Kolesnik and N. Ratanov, Telegraph processes and option pricing, Springer Briefs in Statistics, Springer, Heidelberg, 2013. MR 3115087
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Additional Information
A. O. Pogorui
Affiliation:
Department of Algebra and Geometry, Zhytomyr State University, Velyka Berdychivska St., 40, Zhytomyr, 10008 Ukraine
Email:
pogor@zu.edu.ua
R. M. Rodríguez-Dagnino
Affiliation:
School of Engineering and Sciences, Tecnologico de Monterrey, Av. Eugenio Garza Sada 2501 Sur, C.P. 64849, Monterrey, N.L., México
Email:
rmrodrig@tec.mx
Keywords:
Telegraph process,
random evolutions,
semi-Markov processes,
Erlang distribution,
telegraph equation
Received by editor(s):
December 4, 2018
Published electronically:
January 5, 2021
Article copyright:
© Copyright 2020
American Mathematical Society
