Pathwise estimation of the stochastic functional Kolmogorov-type system

In this paper we stochastically perturb the functional Kolmogorov-type system x ˙ ( t ) = diag ( x 1 ( t ) , … , x n ( t ) ) f ( x t ) into the stochastic functional differential equation d x ( t ) = diag ( x 1 ( t ) , … , x n ( t ) ) [ f ( x t ) d t + g ( x t ) d w ( t ) ] . This paper studies pathwise estimation of the solution to this equation. As the applications, this paper also discusses the pathwise estimation of the solutions of various stochastic Lotka–Volterra-type systems.