G-BESSEL PROCESSES AND RELATED PROPERTIES

doi:10.1007/s10473-026-0220-4

Abstract

Abstract: In this paper, we introduce $ G $-Bessel processes for a class of $ d $-dimensional $ G $-Brownian motions. Under the condition of dimensionality $ d $, we obtain that the $ G $-Bessel process is the solution of the stochastic differential equation. Furthermore, under the stricter condition of dimensionality, we establish the existence and uniqueness of a solution of the stochastic differential equation governing the $ G $-Bessel process and prove the nonattainability of the origin for $ G $-Brownian motion.

Key words: $G$-Brownian motion, $G$-heat equation, $ G $-Bessel process, nonattainability of the origin

CLC Number:

60H10

Mingshang HU, Renxing LI. G-BESSEL PROCESSES AND RELATED PROPERTIES[J].Acta mathematica scientia,Series B, 2026, 46(2): 920-936.

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References

[1] Denis L, Hu M, Peng S. Function spaces and capacity related to a sublinear expectation: Application to $ G $-Brownian motion paths. Potential Anal, 2011, 34: 139-161
[2] Göing-Jaeschke A, Yor M. A survey and some generalizations of Bessel processes. Bernoulli,2003, 9(2): 313-349
[3] Gao F. Pathwise properties and homomorphic flows for stochastic differential equations driven by $ G $-Brownian motion. Stochastic Process Appl, 2009, 119(10): 3356-3382
[4] Hu M, Peng S. On representation theorem of $ G $-expectations and paths of $ G $-Brownian motion. Acta Math Appl Sin Engl Ser, 2009, 25(3): 539-546
[5] Hu M, Ji X, Liu G. Lévy's martingale characterization and reflection principle of $ G $-Brownian motion. J Math Anal Appl,2019, 480(2): Art 123436
[6] Hu M, Ji X, Liu G. On the strong Markov property for stochastic differential equations driven by $ G $-Brownian motion. Stochastic Process Appl, 2021, 131: 417-453
[7] Hu M, Wang F, Zheng G. Quasi-continuous random variables and processes under the $ G $-expectation framework. Stochastic Process Appl, 2016, 126: 2367-2387
[8] Le G J. Brownian Motion, Martingales, and Stochastic Calculus. Berlin: Springe-Verlag, 2016
[9] Karatzas I, Shreve S.Brownian Motion and Stochastic Calculus. New York: Springe-Verlag, 1991
[10] Li X, Peng S. Stopping times and related Itô's calculus with $ G $-Brownian motion. Stochastic Process Appl, 2011, 121: 1492-1508
[11] Lin Q. Some properties of stochastic differential equations driven by the $ G $-Brownian motion. Acta Math Sin Engl Ser, 2013, 29(5): 923-942
[12] Lin Y. Stochastic differential equations driven by $ G $-Brownian motion with reflecting boundary conditions. Electron J Probab, 2013, 18(9): 1-23
[13] Liu G. Exit times for semimartingales under nonlinear expectation. Stochastic Process Appl, 2020, 130(12): 7338-7362
[14] Peng S. Filtration consistent nonlinear expectations and evaluations of contingent claims. Acta Math Appl Sin, 2004, 20: 1-24
[15] Peng S. Nonlinear expectations and nonlinear Markov chains. Chinese Ann Math, 2005, 26B(2): 159-184
[16] Peng S.$ G $-expectation, $ G $-Brownian motion and related stochastic calculus of Itô type//Stochastic Analysis and Applications: Abel Symp 2005. Berlin: Springer, 2007: 541-567
[17] Peng S. Multi-dimensional $ G $-Brownian motion and related stochastic calculus under $ G $-expectation. Stochastic Process Appl, 2008, 118(12): 2223-2253
[18] Peng S.Nonlinear Expectations and Stochastic Calculus Under Uncertainty. Heidelberg: Springer-Verlag, 2019
[19] Ren Y, Hu L. A note on the stochastic differential equations driven by $ G $-Brownian motion. Statist Probab Lett, 2011, 81: 580-585
[20] Revuz D, Yor M.Continuous Martingales and Brownian Motion. Berlin: Springer-Verlag, 1999
[21] Song Y. Properties of hitting times for $ G $-martingales and their applications. Stochastic Process Appl, 2011, 121(8): 1770-1784
[22] Song Y. Characterizations of processes with stationary and independent increments under $ G $-expectation. Ann Inst Henri Poincaré Probab Stat,2013, 49: 252-269
[23] Wang F, Zheng G. Sample path properties of $ G $-Brownian motion. J Math Anal Appl, 2018, 467(1): 421-431
[24] Xu J, Zhang B. Martingale characterization of $ G $-Brownian motion. Stochastic Process Appl, 2009, 119: 232-248
[25] Xu J, Zhang B.Martingale property and capacity under $ G $-framework. Electron J Probab, 2010, 15: 2041-2068
[26] Yor M.Exponential Functionals of Brownian Motion and Related Processes. Berlin: Springe-Verlag, 2001