On the Existence and Uniqueness of Solutions to Rough Fractional Stochastic Differential Equations

Precise modeling of financial asset volatility is significant for robust risk management and derivative pricing‎. ‎Recent scholarly investigations have demonstrated a significant interest in employing stochastic processes with short-term memory for this purpose‎. ‎Consequently‎, ‎rigorous examination of the existence and uniqueness of solutions for these processes assumes critical importance‎. ‎This study commences with the precise definition of a fractional operator for H \in(۰‎, ‎\frac{۱}{۲})‎. ‎Subsequently‎, ‎the finiteness of the second-order moment of the Itô-Skorokhod integral is meticulously investigated‎, ‎utilizing the aforementioned operator‎, ‎specifically within the range of H \in(۰‎, ‎\frac{۱}{۲})‎. ‎Ultimately‎, ‎leveraging this moment and rigorously applying Lipschitz and linear growth conditions‎, ‎and through the application of Gronwall's inequality‎, ‎the existence and uniqueness of solutions for stochastic differential equations with short-term memory are definitively established‎.

کلیدواژه ها:

Fractional Brownian motion‎ ، ‎Wiener integral‎ ، ‎Stochastic differential equations‎ ، ‎Lipschitzian condition ، Gronwall' s inequality