Maximal inequalities for CIR processes
Let \(X\) be a Cox-Ingersoll-Ross process solving the stochastic differential equation \[ dX_t=(a+bX_t)dt+c\sqrt{| X_t| }dB_t \] with \(X_0=0\), where \(a, c > 0\), \(b \in {\mathbb R}\) and \(B\) is a standard Brownian motion started at zero. Some inequalities between the integral functional \(J_{\varphi}(t)=\int_0^t\varphi(X_s)ds\), \(t \geq 0\), and the maximum process \(\sup_{0 \leq s \leq t} X_t\), \(t \geq 0\), where \(x \mapsto \varphi(x)\) is a nonnegative continuous function with some suitable conditions, are derived.
Bessel processes and domination principle
scientific article; zbMATH DE number 2091492