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$ G $-Bessel processes and related properties

Mingshang Hu, Renxing Li

TL;DR

This work extends Bessel process theory to the $G$-expectation framework by constructing $G$-Bessel processes from $d$-dimensional $G$-Brownian motions via the radial process $R_t=|B_t^{x}|$. Under a key condition on $G$ (equivalently a condition on the quadratic variation), the radial process solves a one-dimensional $G'$-SDE driven by a $G'$-Brownian motion, with a drift term $(d-1)/(2R_t)$. The authors establish existence (and, under stronger dimensionality, uniqueness) of solutions in appropriate $G$-Sobolev spaces and prove the quasi-sure nonattainability of the origin, generalizing classical Bessel properties to a nonlinear expectation setting. They further show rotation invariance, connect the model to the linear case, and provide PDE-based capacity estimates to support the stochastic analysis. Overall, the results deepen understanding of radial diffusion under model uncertainty and lay groundwork for applications in finance and nonlinear PDEs.

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.

$ G $-Bessel processes and related properties

TL;DR

This work extends Bessel process theory to the -expectation framework by constructing -Bessel processes from -dimensional -Brownian motions via the radial process . Under a key condition on (equivalently a condition on the quadratic variation), the radial process solves a one-dimensional -SDE driven by a -Brownian motion, with a drift term . The authors establish existence (and, under stronger dimensionality, uniqueness) of solutions in appropriate -Sobolev spaces and prove the quasi-sure nonattainability of the origin, generalizing classical Bessel properties to a nonlinear expectation setting. They further show rotation invariance, connect the model to the linear case, and provide PDE-based capacity estimates to support the stochastic analysis. Overall, the results deepen understanding of radial diffusion under model uncertainty and lay groundwork for applications in finance and nonlinear PDEs.

Abstract

In this paper, we introduce -Bessel processes for a class of -dimensional -Brownian motions. Under the condition of dimensionality , we obtain that the -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 -Bessel process and prove the nonattainability of the origin for -Brownian motion.
Paper Structure (4 sections, 12 theorems, 82 equations)

This paper contains 4 sections, 12 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $G$-Bessel processes
  4. Some properties of $G$-Bessel processes

Key Result

Theorem 2.9

For $t\geq 0$, where $\Gamma$ satisfies (G). Moreover, where $C_{l,Lip}(\mathbb{S}_{d})$ denotes the linear space of all local Lipschitz functions on $\mathbb{S}_{d}$.

Theorems & Definitions (31)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: $G$-normal distribution
  • Definition 2.6: $G$-Brownian motion starting at $x$
  • Remark 2.7
  • Definition 2.8
  • Theorem 2.9
  • Definition 2.10
  • ...and 21 more