Squared Bessel processes under nonlinear expectation
Mingshang Hu, Renxing Li, Xue Zhang
TL;DR
The paper extends squared Bessel process theory to the $G$-expectation setting to incorporate volatility uncertainty. It defines the squared $G$-Bessel process as the square of the modulus of a $G$-Brownian motion, proving it is the unique non-negative solution to the $G$-SDE $Z_t = z + 2\int_{0}^{t} \sqrt{Z_s}\, d\beta_s + d\langle \beta\rangle_t$, and analyzes its path properties in the capacity sense. It provides upper and lower bounds for the Laplace transform $\hat{\mathbb{E}}[\exp(-\lambda Z_t)]$ and establishes a deterministic time-space transformation that yields a $G'$-CIR process, linking squared $G$-Bessel processes to $G'$-CIR dynamics. By connecting nonlinear expectation techniques with classical Bessel and CIR theory, the work enables robust stochastic analysis under volatility ambiguity.
Abstract
In this paper, we define the squared G-Bessel process as the square of the modulus of a class of G-Brownian motions and establish that it is the unique solution to a stochastic differential equation. We then derive several path properties of the squared G-Bessel process, which are more profound in the capacity sense. Furthermore, we provide upper and lower bounds for the Laplace transform of the squared G-Bessel process. Finally, we prove that the time-space transformed squared G-Bessel process is a G'-CIR process.