I am working on projects involving the stochastic approach, and basically Stochastic reaction networks (SRNs) which are continuous-time Markov chains intended to describe, from the kinetic point of view, the time-evolution of chemical systems in which molecules of different chemical species undergo a finite set of reaction channels. These chemical systems are assumed to be at thermal equilibrium, but not necessarily at chemical equilibrium. Stochastic variability has been observed in chemical systems where one or more species are present in low molecule numbers. In those cases, the classical description by reaction-rate ordinary differential equations (ODEs) cannot capture phenomena such as the sudden extinction of one of the species. The theory of SRNs offers instead a natural framework for incorporating the stochastic variability of such systems.

Regarding the scale in which we describe some natural and artificial phenomena, a SRN model corresponds to the microscopic scale, while its reaction-rate ODEs and the Langevin-Itˆo diffusions approximations correspond to its macroscopic and mesoscopic scales.

Attached dissertation of Alvaro Moraes and mainly in chapter 1 you may find the connections between the stochastic description of SRNs and the Langevin diffusions, and the classical reaction-rate ODEs.

Yes, that is the usual way, for deterministic to be a component of stochastic

I believe it's the other way around. Given the stochastic approach, you can construct the deterministic systems. So, if you have a set of chemicals reactions and want to considerer the stochasticity you need to formulate the chemical master equation (SME). SME describes the change of probability distribution over time. Using the derivative of the firsts statistical moment (∑ xd P ( x )/dt ) you can get how the molecular ocunt changes over time.

This tells you that deterministic models are the mean over N stochastic simulations when N tends to infinity. It is important to consider that the stochastic regime works with molecular count, and usually deterministic models with concentrations, so you have to make the properly adecations to the reaction rates or use the avogadro's number.

Not for all the systems this can be done, particularly for nonlinear models are hard to derive the equations.

I believe the word "stochastic" can be losely translated as meaning "random" & I do not think it has a fixed meaning when used in different areas of mathematics. When used in a time series modeling context, I believe the most common useage implies the random component is a basic part of the model & not just a residual random measurement error. Think of a basic stochastic process like a simple random walk. Every step's distance is random in length but it always starts from the end of the last step. Where you stand at any moment is a sum of all the previous random steps. Thus the random term effects both the current value & future values. In stochastic models the random effect defines the forward trajectory while in deterministic models ir does not. So I am not sure how one could convert one to the other because they make radically different & opposing underlying assumptions. Pure stochastic time series models do not have an "error" term, they fundamentally assume that no measurement error exists. There is still uncertainty but it arise in different ways. There are such things as mixed models that have both a deterministic component & a stochastic component but they are not equivalent types of models.

I also recommend these two references (below) that explain the relation between microscopic (scale) model, mesoscopic (scale) model, and the macroscopic (scale) model. The relations there show basically what are the conditions and how to go from one type of modeling to another.

1-Pavliotis, Grigoris, and Andrew Stuart. Multiscale methods: averaging and homogenization. Springer Science & Business Media, 2008.

2- Lachowicz, Mirosław. "Microscopic, mesoscopic and macroscopic descriptions of complex systems." Probabilistic Engineering Mechanics 26.1 (2011): 54-60.