Stochastic processes are very important in statistics and probability, where finding upper and lower bounds of mean-square stochastic integral has led to a basic problem. In this paper we show that for mean-square differentiable stochastic process, the convexity condition in previous well-known results can be replaced by weaker conditions.

Two stage linear models are applicable when the data of some dependent and independent variables was obtained at to time stage, and we want to use from the data of two stage for linear model fitting. In this article we introduce multistage and, as a special case, two-stage linear models. Then we obtain the parameter estimation by two methods and show that the estimation are the same for methods. Since the expression of estimations are very complicated we give some R program for computing the parameter estimation of two-stage linear models, then show its application in an illustrative example. Also we propose a very simple computational methods for parameter estimation which did not need to complicated expression and give and R program for it.

This paper presents new bounds for the left and right fractional mean-square stochastic integrals based on convex stochastic processes. Then a range is proposed that includes a linear combination of the left and right fractional mean-square stochastic integrals. Finally, the previous results presented in this subject are improved.