We study the continuity properties of optimal solutions to stochastic control problems with respect to initial probability measures and applications of these to the robustness of optimal control policies applied to systems with incomplete or incorrect priors. It is shown that for single and multi-stage optimal cost problems, continuity and robustness cannot be established under weak convergence...

Proof. (⇒) Let Ω0 = {ω : limn Xn(ω) = X(ω)}. Suppose P (Ω0) = 1. Let > 0 be given. →∞ Let Am = ∩k=m{|Xk − X| ≤ }. Then Am ⊂ Am+1 ∀m and limm P (Am) = P (∪m=1Am) by →∞ continuity of probability measure. For each ω0, there exists m(ω0) such that |Xk(ω0)−X(ω0)| ≤ for all k ≥ m(ω0). Therefore, ∀ω0 ∈ Ω0, ω0 ∈ Am for some m and we can conclude that Ω0 ⊂ ∪m=1Am and 1 = P (Ω0) ≤ P (∪m=1Am) = limm P (A ...

An incomplete information game is defined by a probability distribution + over a type space and payoff functions u. Probability distribution +$ is strategically close to + if, for any bounded payoff functions u and any equilibrium of the game (+, u), there exists an approximate equilibrium of the game (+$, u) under which all players get approximately the same payoffs. This note shows that two p...

In this work infinitely divisible cylindrical probability measures on arbitrary Banach spaces are introduced. The class of infinitely divisible cylindrical probability measures is described in terms of their characteristics, a characterisation which is not known in general for infinitely divisible Radon measures on Banach spaces. Furthermore, continuity properties and the relation to infinitely...

We do a thorough mathematical study of the notion of full conglomerability, that is, conglomerability with respect to all the partitions of an infinite possibility space, in the sense considered by Peter Walley [1991]. We consider both the cases of precise and imprecise probability (sets of probabilities). We establish relations between conglomerability and countable additivity, continuity, sup...

It is shown that continuity of a family of invariant (Haar) measures on a topological groupoid G is equivalent to the continuity of the implied convolution product f * g for all pairs of functions / and g. An example is given of a groupoid which admits no (continuous) Haar measure. It results, therefore, that the usual C*-algebra associated with a Haar measure on G cannot, in general, be constr...

We present the first result for kernel regression where the procedure adapts locally at a point x to both the unknown local dimension of the metric space X and the unknown Hölder-continuity of the regression function at x. The result holds with high probability simultaneously at all points x in a general metric space X of unknown structure.