Q2. What are the future works in "Robust control and model misspecification" ?
Among other possibilities, this allows the approximating model to miss the serial correlation of exogenous variables and the dynamics of how those exogenous variables impinge on endogenous state variables. This specification keeps the decision maker concerned about models that can be difficult to distinguish from the approximating model from a continuous record of observations on the state vector of a finite length. Via statistical detection error probabilities, Anderson, Hansen, and Sargent ( 2003 ) show how the penalty parameter or the constraint parameter in the robust control problems can be used to identify a set of perturbed models that are difficult to distinguish statistically from the approximating model in light of a continuous record of finite length T of observations on xt.
Q3. How can the authors achieve compactness by adding points to the control set?
By relaxing the linear space structure the authors can achieve compactness by adding points (say the point ∞) to the control set, provided that the authors can extend χ(·, ȟ, x̌) to be upper semicontinuous.
Q4. What is the way to restrict the models entertained at time t?
Taking continuation entropy as a state variable is a convenient way to restrict the models entertained at time t by the minimizing player in the recursive version of constraint game.
Q5. What is the way to restate the benchmark problem?
It is useful to restate the benchmark problem in terms of the probability space that the Brownian motion induces over continuous functions of time, thereby converting it into a nonsequential problem that pushes the state x into the background.
Q6. What is the optimal objective function for the martingale problem?
To analyze outcomes under a sequential timing protocol, the authors think of varying the initial state and define a value function M(x0, z0) as the optimized objective function (28) for the martingale problem.
Q7. How do Hansen and Sargent (2005b) extend the approach of this paper?
By including a hidden state vector and appropriately decomposing the density of next period’s observables conditional on a history of signals, Hansen and Sargent (2005b) extend the approach of this paper to allow a decision maker to have multiple models and to seek robustness to the specification of a prior over them.
Q8. Why does the second equality have to be replaced by a less than or equal sign?
Because minimization occurs first, without the assumption the second equality would have to be replaced by a less than or equal sign ( ≤).
Q9. What is the recursive version of the constraint problem?
By modifying the set of priors over time, constraint problem (4) states a recursive version of that nonsequential constraint problem.
Q10. What is the risk sensitive interpretation of the martingale?
The risk sensitive interpretation excludes worries about misspecified dynamics and instead enhances the control objective with aversion to risk in a way captured by the local variance of the continuation value.
Q11. What is the relative entropy of q?
Relative entropy is well known to be convex in the probability measure q̃ (e.g. see Dupuis and Ellis (1997)), and hence R̃ is convex in q.
Q12. Why do the authors introduce discounting in part?
The authors introduce discounting in part to provide an alternative interpretation of the recursive formulation of risk-sensitive control as expressing a fear of model misspecification rather than extra aversion to well understood risks.
Q13. How can the penalty parameter be used to identify a set of alternative models?
Via statistical detection error probabilities, Anderson, Hansen, and Sargent (2003) show how the penalty parameter or the constraint parameter in the robust control problems can be used to identify a set of perturbed models that are difficult to distinguish statistically from the approximating model in light of a continuous record of finite length T of observations on xt.
Q14. How did Jacobson and Whittle show that the risk-sensitive control law can be computed?
Jacobson (1973) and Whittle (1981) first showed that the risk-sensitive control law can be computed by solving a robust penalty problem of the type the authors have studied here, but without discounting.
Q15. What is the advantage of working with the induced distributions?
An advantage of working with the induced distributions is that a convexity property that helps to establish the connection between the two games is easy to demonstrate.
Q16. What is the contribution to entropy coming from the distortion of the probabilities?
The contribution to entropy coming from the distortionof the probabilities is the discrete state analogue of ∫ log (dqt dq0t) dqt, namely,I(pt) = pt log pt + (1− pt) log(1− pt) + log 2.
Q17. What did Hansen and Sargent (1995) show how to introduce discounting and still preserve?
Hansen and Sargent (1995) showed how to introduce discounting and still preserve much of the mathematical structure for the linear-quadratic, Gaussian risk-sensitive control problem.
Q18. What is the reason why the decision maker would reject multiple priors?
If multiple priors truly are a statement of a decision maker’s subjective beliefs, the authors think it is not appropriate to dismiss such beliefs on the grounds of dynamic inconsistency.