Thank you very much, Daniel. I appreciate your insights, and I believe they are particularly suitable for my paper. Indeed the presented methodology involves several steps: 1) defining the forward transition matrix; 2) ensuring that the forward transition matrix satisfies a set of constraints (in C++, we could define the abstract concept of linear requirements for transition matrix and then derive concrete classes); next, applying the simplex method to the overall conditions, resulting in two forward transition matrices solutions: one that minimizes the target (e.g. an exotic payoff) and one that maximizes it. For each solution, we can calculate all the numbers for all contracts (plain vanilla contracts, of course, remain the same) beyond just the target. All of this should be presented as a block diagram with a clear workflow. In this way, it should be easier to explain each block mathematically. And following this structure, the paper writing will follow a clear path arriving to the final point: getting narrow bounds for exotics respect to the set of all well formed pricing models. Discard all not in line considerations that do not have a representation in the block diagram . Have I understood your point correctly? 

Dispate the not very good writing and paper structuring, may I ask your overall impression?

Thanks for giving me the opportunity to interact with you
Marco

Re "can always be integrated out". 

It's not clear to me what is being asserted. 

It's true that, in Hidden Markov Models, for example, there is indeed a lot of integrating out. I have a whole chapter on "Stochastic Volatility as a Hidden Markov Model" in my "Option Valuation under Stochastic Volatility II" book. 

Separately, Dupire/Gyongy theory does a lot of integrating out of univariate or multivariate stochastic volatility in the construction of a local volatility surface for diffusions. 

However, there remains the problem of "re-calibration". 

In any event, if you can make/prove a careful statement about the application of your theory to data generating processes with additional state variables (hidden or not), my suggestion is to include that in your paper.    

1. First, you have to understand the problem. (can you pose it as a block diagram as well?)
2. After understanding, make a plan.(algorithm) ==> Data Flow Diagram DFD is super
3. Carry out the plan. C++ 
4. Look back on your work. How could it be better?

Is this article by Prof Madan relevant to Marco's article?

1. Take two future dates and calculate, for that specific model (be it a stochastic volatility model, a stochastic local volatility model, or a jump-diffusion process), the joint probabilities of being at S1 at time T1 and S2 at time T2. This calculation must be done independently from the value of any other auxiliary variables, such as the number of jumps or stochastic volatility. It should be just the joint probability for the underlying.
2. These joint probabilities, forming a matrix of numbers, essentially represent the transition matrix over two dates, and one should verifies that satisfy the chosen axioms or rules to declare a model a good one.
3. If this is confirmed, the model is included in the set of treated models because a transition matrix for that model has been calculated (e.g., via Monte Carlo), and this matrix satisfies the rules/axioms of a good model. At that point, the simplex method ensures that using this model to calculate the price of a derivative will fall within the MIN/MAX band. I have conducted this verification, for example, in the case of a jump-diffusion model, where an additional variable is the number of jumps. Nevertheless, I agree that it is a point to carefully explore in a concrete example, as Alan suggests.