Fractional Cox–Ingersoll–Ross process with non-zero «mean»

Yuliya Mishura, Anton Yurchenko-Tytarenko
2018 Modern Stochastics: Theory and Applications  
In this paper we define the fractional Cox-Ingersoll-Ross process as X_t:=Y_t^21_{t<{s>0:Y_s=0}}, where the process Y={Y_t,t>0} satisfies the SDE of the form dY_t=1/2(k/Y_t-aY_t)dt+σ/2dB_t^H, {B^H_t,t>0} is a fractional Brownian motion with an arbitrary Hurst parameter H∈(0,1). We prove that X_t satisfies the stochastic differential equation of the form dX_t=(k-aX_t)dt+σ√(X_t)∘ dB_t^H, where the integral with respect to fractional Brownian motion is considered as the pathwise Stratonovich
more » ... al. We also show that for k>0, H>1/2 the process is strictly positive and never hits zero, so that actually X_t=Y_t^2. Finally, we prove that in the case of H<1/2 the probability of not hitting zero on any fixed finite interval by the fractional Cox-Ingersoll-Ross process tends to 1 as k→∞.
doi:10.15559/18-vmsta97 fatcat:7caep5la3ngmvcjnnhmx4rwudq