Existence theory for a stochastic differential equation

TLDR: In this article, a non-linear function stochastic differential equation was studied and sufficient conditions were developed for the existence of random solutions, second order Stochastic processes, for the above equation and to place bounds upon these random solutions.

Abstract: A non-linear function stochastic differential equation was studied where t𝛆R+={t;t ⩾ 0},ω𝛆 Ω, Ω being the underlying sot of a complete probability measure space ( Ω,A,P) The random process x(t;ω) is the unknown stochastic function defined on R+ × Ω h(t, x;ω ) ) is the stochastic term defined for t𝛆 R+ and x ( t;ω)eG(a Branch Space); and n(t, x ω) is a random variable defined for te R+ω 𝛆 Ω, and x 𝛆 F{Grcub; (a Frcchet space). The purpose of this paper is to develop sufficient conditions for the existence of random solutions, second order stochastic processes, for the above equation and to place bounds upon these random solutions. Several examples are also presented which illustrate the usefulness of the theoretical findings.