### A generalization of Osgood's test and a comparison criterion for integral equations with noise.

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We consider the stochastic differential equation ${X}_{t}=X\u2080+{\int}_{0}^{t}({A}_{s}+{B}_{s}{X}_{s})ds+{\int}_{0}^{t}{C}_{s}d{Y}_{s}$, where ${A}_{t}$, ${B}_{t}$, ${C}_{t}$ are nonrandom continuous functions of t, X₀ is an initial random variable, $Y=({Y}_{t},t\ge 0)$ is a Gaussian process and X₀, Y are independent. We give the form of the solution (${X}_{t}$) to (0.1) and then basing on the results of Plucińska [Teor. Veroyatnost. i Primenen. 25 (1980)] we prove that (${X}_{t}$) is a quasi-diffusion proces.

Let (Ω,,P) be a probability space and let τ: ℝ×Ω → ℝ be a function which is strictly increasing and continuous with respect to the first variable, measurable with respect to the second variable. Given the set of all continuous probability distribution solutions of the equation $F\left(x\right)={\int}_{\Omega}F\left(\tau (x,\omega )\right)dP\left(\omega \right)$ we determine the set of all its probability distribution solutions.