Generalisation of fractional Cox–Ingersoll–Ross process

A Fractional Heston-Type Model as a Singular Stochastic Equation Driven by Fractional Brownian Motion

“…From this expression, we may also conclude that |Z ε t | p ≤ C, where C = C(C 1 , C 2 ) and where C 1 and C 2 are a non-random constants defined by ( 22) and (23), respectively. This shows that E Z ε t p < ∞ and consequently, E sup t∈[0,T] Z t p < ∞.…”

“…Proof. Here, we highlight the proof of this theorem by contradiction, and we refer the reader to Mpanda et al [23] for a complete and comprehensive proof. Let τ(ω) = inf{t ≥ 0 : Z t (ω) = 0} be the first time that the process (Z t ) t∈[0,T] hits zero and τ ε (ω) = sup{t ∈ (0, τ(ω)) : Z t (ω) = ε} be the last time (Z t ) hits ε before reaching zero.…”

“…where σ(Y t ) represents the volatility of the infinitesimal log-return dX t := dS t /S t with (Y t ) t≥0 a fractional Cox-Ingersoll-Ross (fCIR) process that captures both long and short-range dependency. We adopt the definition provided by Mishura and Yurchenko-Tytarenko [22] or Mpanda et al [23] and describe the stochastic process (Y t ) t≥0 as…”