Document Type : Research Article
Authors
1 Department of Applied Mathematics, Faculty of Mathematical Science, University of Guilan
2 Mathematical Science, Applied Mathematics
Abstract
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 $H \in(0, \frac{1}{2})$. 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 $H \in(0, \frac{1}{2})$. 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.
Keywords
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