Veranstaltungen

Similar to ODEs, time-continuous control systems have trajectory-wise time rescaled solutions when a state-dependent scalar field is multiplied with the vector field of the control system and the time variable of the vector field is rescaled appropriately. In my talk, I will give an introduction about control systems, controllability and optimal control. I will explain how known results about controllability like the Kalman criterion can be extended by a suitable choice of the control function in combination with trajectory-wise time rescaling. This strategy has its limits with regard to Chow's theorem1 for affine nonlinear control systems. Finally, I will present how optimal control problems with a fixed time domain change under trajectory-wise time rescaling and that the new optimizer and the new adjoint solution of the necessary conditions from Pontryagin's maximum principle² are time-rescaled versions of the original optimizer and adjoint solution. [1] D. Cheng, X. Hu, and T. Shen. Analysis and Design of Nonlinear Control Systems. Springer Berlin, Heidelberg, 2011. Theorem 3.5 [2] R. Vinter. Optimal Control. Birkhäuser Boston, MA, 2010. Theorem 8.7.1

We extend the scope of a recently introduced dependence coefficient between a scalar response Y and a multivariate covariate X to the case where X takes values in a general metric space. Particular attention is paid to the case where X is a curve. While on the population level, this extension is straight forward, the asymptotic behavior of the estimator we consider is delicate. It crucially depends on the nearest neighbor structure of the infinite-dimensional covariate sample, where deterministic bounds on the degrees of the nearest neighbor graphs available in multivariate settings do no longer exist. The main contribution of this paper is to give some insight into this matter and to advise a way how to overcome the problem for our purposes. As an important application of our results, we consider an independence test.

We extend the popular Gibson and Schwartz (1990) and Schwartz and Smith (2000) two-factor models for the spot price of a commodity to include stochastic volatility and correlation. This generalization is based on the Wishart variance-covariance matrix process. For both of the extended models we present the joint characteristic functions of the two state variables. The original models are known to fit the term-structure of implied volatility in futures and options markets very well. However, the extended models are also able to match volatility smiles observed in these markets. Regarding the analysis of financial time series, the assumption of a constant correlation between the state variables is known to be too restrictive. Introducing time-varying correlation via the Wishart process allows us to study its empirical behaviour in commodity markets through the use of filtering techniques.

We study causality in systems that allow for feedback loops among the variables via models of cross-sectional data from a dynamical system. Specifically, we consider the set of distributions which appears as the steady-state distributions of a stochastic differential equation (SDE) where the drift matrix is parametrized by a directed graph. The nth-order cumulant of the steady state distribution satisfies the corresponding nth-order continuous Lyapunov equation. Under the assumption that the driving Lévy process of the SDE is not a Brownian motion (so the steady state distribution is non-Gaussian) and the coordinates are independent, we are able to prove generic identifiability for any connected graph from the second and third-order Lyapunov equations while allowing the cumulants of the driving process to be unknown diagonal. We propose a minimum distance estimator of the drift matrix, which we are able to prove is consistent and asymptotically normal by utilizing the identifiability result.