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Maximal Inequalities for CIR Processes

Authors:

Institution:

(1) Department of Mathematics, College of Science, Donghua University, 1882 West Yan'an Rd., Shanghai, 200051, P.R. China;(2) Mudanjiang Joint Professional University, Mudanjiang, 157000, P.R. China

Abstract:

Let X be a Cox—Ingersoll—Ross (CIR) process given by

${\text{d}}X_t = \left( {a + bX_t } \right){\text{d}}t + c\sqrt {\left| {X_t } \right|} {\text{d}}B_t$

with X ₀ = 0, where a, c>0, $b \in \mathbb{R}$ and B a standard Brownian motion starting at zero. We obtain some inequalities between the integral functional $J_\varphi \left( t \right) = \int_0^t {\varphi \left( {X_s } \right){\text{d}}s,t \geqslant 0}$ , t ⩾ 0 and the maximal process sup_0⩽s⩽t X _s, t ⩾ 0, where x↦ϕ(x) a nonnegative continuous function with some suitable conditions.

Keywords:

Bessel processes and domination principle Cox— Ingersoll— Ross model diffusion processes It?'s formula maximal inequalities

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