Viability for mixed stochastic differential equations driven by fractional Brownian motion and its application

ABSTRACT

In this paper, we focus on a class of stochastic differential equations driven by the standard Brownian motion and fractional Brownian motion with Hurst parameter $1/2<H<1$ under a weaker conditions than Lipschitz one. In the sense of the pathwise Riemann–Stieltjes integral, we prove the convergence of solutions for the considered equations. By making use of some transformation techniques and approximation means, we obtain some sufficient conditions on the viability for the stochastic systems under investigation. Subsequently, by using some distance functions, we establish some necessary conditions and some sufficient conditions for the viability property with respect to a given non-empty closed set K, but K can be not smooth. As applications of the results, we explore the comparison theorems about the mixed stochastic differential equations driven by fractional Brownian motion under some non-Lipschitz conditions.

KEYWORDS:

• Viability
• non-Lipschitz condition
• fractional Brownian motion
• mixed SDEs
• comparison theorem

2000 MATHEMATICS SUBJECT CLASSIFICATION(S):

• 60H10

Data availability statement

No data, models, or code were generated or used during the study.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is partially supported by the NNSF of China [No. 11901058]. Hubei Provincial Natural Science Foundation of China [2021CFB543].