[Robust Control and Model Uncertainty], Belirsizlik Modeli ve Sağlamlılık Kontrolü

Belirsizlik Modelive Sağlamlılık Kontrol

On the equivalence of probability spaces

For a general class of Gaussian processes $W$, indexed by a sigma-algebra $\mathscr F$ of a general measure space $(M,\mathscr F, \sigma)$, we give necessary and sufficient conditions for the validity of a quadratic variation representation for such Gaussian processes, thus recovering $\sigma(A)$, for $A\in\mathscr F$, as a quadratic variation of $W$ over $A$. We further provide a harmonic analysis representation for this general class of processes. We apply these two results to: $(i)$ a computation of generalized Ito-integrals; and $(ii)$ a proof of an explicit, and measure-theoretic equivalence formula, realizing an equivalence between the two approaches to Gaussian processes, one where the choice of sample space is the traditional path-space, and the other where it is Schwartz' space of tempered distributions.Comment: To appear in Journal of Theoretical Probabilit

Certainty equivalence and model uncertainty

Simon’s and Theil’s certainty equivalence property justifies a convenient algorithm for solving dynamic programming problems with quadratic objectives and linear transition laws: first, optimize under perfect foresight, then substitute optimal forecasts for unknown future values. A similar decomposition into separate optimization and forecasting steps prevails when a decision maker wants a decision rule that is robust to model misspecification. Concerns about model misspecification leave the first step of the algorithm intact and affect only the second step of forecasting the future. The decision maker attains robustness by making forecasts with a distorted model that twists probabilities relative to his approximating model. The appropriate twisting emerges from a two-player zero-sum dynamic game.

Efficiency versus Robustness of Markets - Why improving market efficiency should not be the only objective of market regulation

The efficiency of capital markets has been questioned almost as long as the efficient market hypothesis had been worked out. Numerous critics have been formulated against this hypothesis, questioning notably the behavioural assumptions underlying the efficient market hypothesis. The present contribution does not focus on the behavioural assumptions but rather looks at the implications of focusing purely on the objective of market efficiency when considering market design questions. Hence it aims at discussing the following, possibly rather fundamental issue: Is the objective of efficiency, which has guided most of the market reforms in the last decades, sufficient? Or has it to be complemented by the objective of robustness? Mathematical and engineering control theory has developed the concept of robust control (e.g. Zhou and Doyle, 1998) and it has been shown that there is always a trade-off between the efficiency of a control system and its robustness (cf. e.g. Safonov, 1981, Doyle et al., 1988). The efficiency of the system describes its reactions to disturbance signals. The lower the integral loss function over the so-called transfer or sensitivity function, the less a system is affected by disturbances such as demand fluctuations, and the more efficient is the control. The economic equivalent clearly is the maximisation of welfare, which results in an efficient economic system. Robustness by contrast is defined as stability of the control system in the presence of model uncertainty (deviations in the model parameters or misperceptions of the underlying system). These concepts are applied to the financial markets in their interaction with the real economy. The financial markets being understood as the controllers of real world activity through investments, the implications of misperceptions in the financial sphere are analysed both theoretically and in an application example. From the theory it may readily derived that financial markets providing efficient, i.e. welfare-optimal solutions, must have limitations with respect to robustness. Also in the application example it turns out that in the presence of potential misperception a reduction of irreversible cost shares in investments may lead to an increase in overall expected system costs. Hence improvements in (conventional) market efficiency may be counter-productive by facilitating misallocation of capital as a consequence of misperceptions in the financial markets. This leads to the conclusion that a sole focus on the efficiency objective in market design is problematic and some of the recent turmoil in financial markets may be explained by the lack of consideration given to robustness issues.market efficiency, robustness, optimal control, stochastic dynamic growth

Acknowledgement Misspecification in Macroeconomic Theory

We explore methods for confronting model misspecification in macroeconomics. We construct dynamic equilibria in which private agents and policy makers recognize that models are approximations. We explore two generalizations of rational expectations equilibria. In one of these equilibria, decision makers use dynamic evolution equations that are imperfect statistical approximations, and in the other misspecification is impossible to detect even from infinite samples of time-series data. In the first of these equilibria, decision rules are tailored to be robust to the allowable statistical discrepancies. Using frequency domain methods, we show that robust decision makers treat model misspecification like time-series econometricians.