Indeed: By definition of the penalty and our considerations in Section 4.8.1, ˆ
αmint (θ) → 0 as t → ∞ for all θ ∈ Θ. Secondly, as the maximum likelihood estimator is asymptotically stable, i.e. ˆθt → θ¯0, the conditional reference distributionsPθˆ(·|Ft) converge. Thus, the worst case instantaneous distribu-tionsPθ
∗
t converge as in Definition 4.7.1 due to continuity of the entropy and as the effective domain of the penalty is given by conditional distributions, a fact that is made particularly precise in [Maccheroni et al., 06b].7
measure ρ∞ in that case.
By virtue of a counterexample, we have shown necessity of continuity from below for our result. However, we have shown that time-consistency is not necessary for the result to hold. In particular, we have obtained a more general existence result for the limiting risk measure ρ∞ than in [F¨ollmer & Penner, 06]. Our generalization of the Blackwell-Dubins theorem was shown to be equivalent to the notion of the parameter being eventually learned upon and the notion of asymptotic precision in [F¨ollmer & Penner, 06]
in the time-consistent case.
Further research should be conducted in the direction of our results.
First, of course, the riddle of explicitly constructing convex risk measures by virtue of the penalty function is still to solve; in particular, how a learn-ing mechanism might be introduced without destroylearn-ing the assumption of time-consistency. Weaker notions of time-consistency that are satisfied in a “learning” environment should be introduced along with a comprehensive theory allowing for solutions of tangible economic and social problems.
In the article at hand, we have considered risky projects with final payoffs, i.e. random variables of the formX ∈ F. We have shown convergence of con-vex risk measures to the conditional expected value with respect to the true underlying distribution: a generalization of the Blackwell-Dubins theorem to (not necessarily time-consistent) convex risk measures for final payoffs. To us it seems being an interesting, yet challenging, task to generalize our result to the case of convex risk measures for stochastic payoff processes (Xt)t with respect to some filtration (Ft)t, where eachXt denotes the stochastic payoff in periodt. [Cheridito et al, 06] introduce dynamic convex risk measures for these stochastic processes and elaborately discuss time-consistency issues but do not inspect limiting behavior. A major difficulty in the case of stochastic processes is that the assumption of equivalent distributions should be re-placed by local equivalence, cp. [Riedel, 09]. Hence, the main question turns out to be if the result still holds assuming local instead of global equivalence
as done here.
Chapter 5
Closing Remarks
Within the three essays of this thesis we have tackled several problems arising in case of dynamic coherent as well as convex risk measures or, equivalently, dynamic variational preferences. Each essay is elaborately given in one chap-ter and finalized by a conclusion stating achievements of that essay’s results as well as limitations and ideas for further research. Nevertheless, we briefly summarize our results here at the very end:
First, we have generalized the Best-Choice or Secretary problem to the case of an ambiguous number of applicants. For this problem we have achieved a result on the number of stopping islands generalizing the main theorem in [Presman & Sonin, 72]. In order to achieve this, we have encoun-tered several problems in directly generalizing the risky to the ambiguous problem and hence have built a model in terms of assessments.
Thereafter, we have build a general theory for optimal stopping of pay-off processes in context of time-consistent dynamic variational preferences.
In order to achieve our results on optimal stopping times by virtue of so called variational Snell envelopes extending [Riedel, 09], we have introduced the notion of variational supermartingales and have built an accompanying martingale theory. We have applied our insights to dynamic entropic risk measures and average value at risk.
In the third article, we have considered dynamic convex risk measures when information is gathered in course of time. We have generalized the fundamental Blackwell-Dubins theorem from [Blackwell & Dubins, 62] to not necessarily time-consistent dynamic convex risk measures and have thus shown their convergence to conditional expected values with respect to the true un-derlying distribution: Intuitively the result shows that uncertainty vanishes but risk endures.
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