On the existence and uniqueness of solutions to natural equations with non-Lipschitz conditions

Abstract In this paper, we concentrate on a prominent financial model known as the natural model. Specifically, this model is represented by a stochastic differential equation called the natural equation, which plays a crucial role in our investigation. More precisely, our primary aim is to establish the existence and uniqueness of the solution to the natural equation under a condition weaker than the Lipschitz condition, termed the non-Lipschitz condition. Our investigation is grounded in certain inequalities, such as Gronwall’s inequality, Bihari’s inequality, and Burkholder–Davis–Gundy (BDG) inequality. Additionally, the mean square stability of the solutions to the natural equation is obtained in this research.