The Existence and Uniqueness of Solutions for Mean-Reverting <i>γ</i>-Process

Abstract

Empirical studies show that more and more short-term rate models in capturing the dynamics cannot be described by those classic ones. So the mean-reverting γ -process was correspondingly proposed. In most cases, its coefficients do not satisfy the linear growth condition; even they satisfy the local Lipschitz condition. So we still cannot examine its existence of solutions by traditional techniques. This paper overcomes these difficulties. Firstly, through using the function Lyapunov, it has proven the existence and uniqueness of solutions for mean-reverting γ -process when the parameter . Secondly, when , it proves the solution is non-negative. Finally, it proves that there is a weak solution to the mean-reverting γ -process and the solution satisfies the track uniqueness by defining a function ρ . Therefore, the mean-reverting γ -process has the unique solution.