Systems of differential inequalities and stochastic differential equations. III

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Abstract
Consider the system of stochastic differential equations x′ (t,ω) =f (t,x (t,ω),ω), x (t0,ω) =x0(ω), where f (t,x (t,ω),ω) is a product measurable n‐dimensional random vector function whenever x (t,ω) is a product measurable random function, and it satisfies the desired regularity conditions to ensure the existence of solution process. By developing systems of random differential inequalities, a very general comparison theorem in the framework of a vector Lyapunov function is developed, and furthermore sufficient conditions are given for the stability of solutions in probability, in the mean and with probability one.

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