The Existence and Uniqueness of SDE Solutions Diriged by Multiractional Brownian Motion , Based on Several Developed Methods

To overcome some of the limitations of traditional fractional Brownian motion (fBm), multifractional Brownian motion (mBm) was created. The Holder exponent of mBm can vary along the trajectory, unlike fBm, which is advantageous for modeling processes whose regularity varies over time, like internet traffic or photos. This is the main difference between the two processes. Many continuous observations can be modeled by stochastic differential equations governed by mBm, especially in biology and finance. Using a non-stationary multifractional Brownian motion with a time-dependent Hurst parameter as the noise source and a simply linear drift coefficient, we first show the existence and uniqueness of a solution to stochastic differential equations in this article. Next, we examine multifractional Brownian motion (mBm) driven stochastic differential equation models that allow for the simulation of some discontinuity-containing phenomena.