A note on the continuity for Caputo fractional stochastic differential equations

Area of Science:

• Mathematics
• Stochastic Analysis
• Fractional Calculus

Background:

• Stochastic differential equations (SDEs) are fundamental in modeling complex systems.
• Fractional calculus extends the concept of differentiation and integration to non-integer orders.
• Caputo fractional derivatives are widely used due to their physical interpretability.

Purpose of the Study:

• To establish the well-posedness (global existence and uniqueness of solutions) for Caputo fractional stochastic differential equations.
• To investigate the continuity of solutions with respect to the fractional order parameter.

Main Methods:

• Utilizing techniques from stochastic analysis and fractional calculus.
• Applying fixed-point theorems to demonstrate existence and uniqueness.
• Analyzing solution behavior under variations in the fractional order.

Main Results:

• The global existence and uniqueness of solutions are proven under conditions analogous to integer-order SDEs.
• The continuity of solutions with respect to the fractional order is established.

Conclusions:

• The theoretical framework for Caputo fractional SDEs is strengthened.
• The findings provide a foundation for further research in fractional stochastic modeling.