Finite-Time Stability of Linear Stochastic Proportional Fractional-Order Systems with Time Delay

This paper studies the finite-time stability of linear stochastic fractional-order systems with time delay, in which the dynamics are generated by the proportional Caputo fractional derivative of order ς∈(12,1) with proportional parameter ϖ∈(0,1]. Starting from the integral representation of solutions and using the Itô isometry, we derive a mean-square estimate in which the exponential factor of the proportional kernel sharpens the constants. Combining this estimate with the classical Gronwall inequality and with a generalized one, we establish two finite-time stability criteria: an iterative criterion built on the delay intervals and a closed-form criterion of the Mittag–Leffler type. For ϖ=1, both criteria reduce exactly to known results for Caputo-type stochastic delay systems, so the present results extend the theory to the whole proportional family. The two criteria are compared analytically, a weighted refinement that reduces their conservatism is given, and the influence of the memory of the fractional operator and of the proportional parameter is discussed. Numerical simulations, based on a Monte Carlo scheme for the integral representation of the solution with quantified sampling error, illustrate the two criteria, compare their certified stability horizons, and confirm the theoretical findings.