On the solutions of Conformable Stochastic Differential Equations

Abstract

In this study, we investigate the solution properties of conformable stochastic differential equations, with a fractional order $\alpha\in (1/2,1)$ in the context of spaces $\mathcal{L}^q(\Omega, \mathcal{F}_t,\mathbb{P})$, $q \geq2$. Under some assumptions on the drift and diffusion terms, including Lipschitz continuity and essential boundedness, we derive four main results:  Firstly, we establish the existence and uniqueness of solutions. Secondly, we show the continuous dependence of solutions on the initial values. Thirdly, we show that the solutions possess H\"older continuous regularity. Lastly, we demonstrate the continuous dependence of solutions on the fractional order. To prove these results, we employ a variety of analytical techniques from stochastic calculus and fractional analysis. In particular, we utilize the Gr\"onwall inequality as well as the Burkholder-Davis-Gundy inequality.