This paper develops a simultaneous model and moment selection procedure for factor copula models. Since the density of the factor copula is generally not known in closed form, widely used likelihood or moment based model selection criteria cannot be directly applied on factor copulas. The new approach is inspired by the methods for GMM proposed by Andrews (1999) and Andrews & Lu (2001). The consistency of the procedure is proved and Monte Carlo simulations show its good performance in finite samples in different scenarios of sample sizes and dimensions. The impact of the choice of moments in selected regions of the support on model selection and Value-at-Risk prediction are further examined by simulation and an application to a portfolio consisting of ten stocks in the DAX30 index.
In this talk we will formally show that the continuum counterpart of the discrete individual-based mechanical model that describes the dynamics of two contiguous cell populations with different proliferative and mechanical characteristics is given by a free-boundary problem for the cell densities. We will show the well-posedness result for the free-boundary problem and construct travelling-wave solutions. Numerical simulations results will be used to demonstrate an excellent agreement between numerical solutions of the individual-based model and of the corresponding free-boundary problem and the travelling-wave analysis.
In this paper, we propose a new approach for quantifying portfolio diversification. The proposed framework defines diversification with respect to a risk measure, and benefits can be interpreted as arising from risk diversification. When satisfying the axioms of coherency, the derived functional is called a coherent risk diversi€fiation measure, and it quantifies the percentage of idiosyncratic risk diversified in a portfolio. We also show that under the assumption of elliptically distributed returns, all coherent risk measures depend only on the first two moments of asset returns and portfolio distributions. Finally, we test the proposed risk diversification measures in empirical applications, taking into account various levels of risk aversion. We discuss the concept of mean-risk-diversification efficient frontiers and illustrate how risk-diversified portfolios perform during periods of financial distress. We further examine the ability of portfolio strategies based on risk diversification to outperform given tangent portfolios.
A proper investment of capital is one of the most important decisions financial companies make. There are many ingredients contributing to an ongoing escalation of complexity for investors: an increase in stochastic (latent) factors driving market prices, the subsequent boost in uncertainty, a strengthening of international and local regulations, and the expansion of customized investment products, just to mention a few. This presentation describes some of these challenges using a combination of mathematical and financial concepts, including but not limited to: expected utility theory, continuous-time processes, optimal control, robust analysis and constrained portfolio optimization. Closed-form solutions to some of these problems are presented with an emphasis on their financial implications and benefits on a comparison to existing, suboptimal practices. In this context, I highlight some open problems and promising new approaches.
In this talk we will explore the gamma calculus for Markov semigroups. We will introduce the Bakry-Émery curvature condition and deduce gradient bounds, Poincaré inequalities and log-Sobolev inequalities. The goal is to show exponential fast convergence to an equilibrium. We want to apply the same in the theory of PDEs. It turns out that the curvature condition is not fulfilled by the kinetic Fokker-Planck equality. Therefore we introduce a new calculus and a new curvature condition.
We provide uniform extension and trace operators for $W^{1,p}$ -functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. Such extension and trace operators are important for compactness in stochastic homogenization. In contrast to former results, we use very weak assumptions on the geometry which we call local $(\delta,M)$-regularity and isotropic cone mixing. The first concept measures local Lipschitz regularity of the domain while the second measures the mesoscopic distribution of void space. In particular we do not require a minimal distance between the inclusions and we allow for globally unbounded Lipschitz constants and percolating holes. A typical example is the Poisson ball process, i.e. an i.i.d. set of balls which are allowed to intersect.
This work explores some properties of the random connection model. Using modern results in continuum percolation, we give a proof of the equality of the percolation critical value and the susceptibility critical value by adapting a method of Aizenman and Barsky from 1987. This demonstrates a result of Meester in 1987 anew and in a more straightforward way. Further, we prove the sharp phase transition for the random connection model under the assumption that its connection function has finite support. Here, we utilize a method developed by Duminil-Copin, Raoufi and Tassion. Finally, we expose and discuss some difficulties when applying the method to random connection models with other connection functions, for instance, exponentially decaying ones.
The aim of this talk is to present some recent results on a relaxation of multi-marginal Kantorovich optimal transport problems with a view to the design of numerical schemes to approximate the exact optimal transport problem. More precisely, the approximate problem considered in this talk consists in relaxing the marginal constraints into a finite number of moment constraints, while the state space remains unchanged (typically a subset of R^d for some positive integer d). Using Tchakhaloff’s theorem, it is possible to prove the existence of minimizers of this relaxed problem and characterize them as discrete measures charging a number of points which scales at most linearly with the number of marginals in the problem. In the particular case of a symmetric multi-marginal problem, like the Coulomb cost optimal transport problem which is the semi-classical limit of the Lévy-Lieb functional [1], the number of points charged by minimizers is independent of the number of electrons, thus avoiding the curse of dimensionality. This result is strongly linked to the work [2] and opens the way to the design of new numerical schemes exploiting the structure of these minimizers. Some preliminary numerical results exploiting this structure will be presented.
[1] COTAR, Codina, FRIESECKE, Gero, et KLÜPPELBERG, Claudia. Smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the Hohenberg–Kohn functional. Archive for Rational Mechanics and Analysis, 2018, vol. 228, no 3,, p.891-922. [2] FRIESECKE, Gero et VÖGLER, Daniela. Breaking the curse of dimension in multi-marginal kantorovich optimal transport on finite state spaces. SIAM Journal on Mathematical Analysis, 2018, vol. 50, no 4, p. 3996-4019.
In this compact course, the four lectures from Monday, 20th to Thursday 23rd January will cover the following topics:
Lecture 1: Global Complexity Estimates for the Second-Order Methods
Lecture 2: Accelerated second-order methods. Lower complexity bounds
Lecture 3: Universal second order methods
Lecture 4: Implementable Tensor Methods
For all dates, detailed information and to register, please visit the IGDK Website: https://igdk1754.ma.tum.de/IGDK1754/CC_Nesterov
At the end of the 1990s it was discovered by Jordan/Kinderlehrer/Otto that the diffusion equation is a gradient flow in the space of probability measures, where the driving functional is the Boltzmann-Shannon entropy, and the dissipation mechanism is given by the 2-Wasserstein metric from optimal transport. This result has been the starting point for striking developments at the interface of analysis, probability, and metric geometry.
In this talk I will review joint work with Eric Carlen, in which we introduced new optimal transport metrics that yield gradient flow descriptions for dissipative quantum systems with detailed balance. This approach yields functional inequalities related to convergence to equilibrium in several examples.
When one consider a non-linear system x'=F(x), the local behavior at an equilibrium point (say 0), is given by the sign of the real parts of the eigenvalues of the Jacobian matrix of F at 0. The aim of this talk is to give similar results when X is now a stochastic process that randomly switches between several vector fields having 0 as an equilibrium. Briefly put, our main result is that the long term behaviour of the process is determined by the behaviour of the process obtained by linearisation at the origin, which is itself given by the sign of the top Lyapunov exponent (multiplicative ergodic theorem). We provide several examples coming from epidemiological models, Lotka-Volterra model and Lorenz vector field. This talk is based on a joint work with Michel Benaïm.
Stationary non equilibrium states are characterized by the presence of steady currents flowing through the system as a response to external forces. We model this process considering the simple exclusion process in one space dimension with appropriate boundary mechanisms which create particles on the one side and kill particles on the other. The system is designed to model Fick's law which relates the current to the density gradient. In this talk we focus on the fluctuations around the hydrodynamic limit of the system. The main technical difficulty lies on controlling the correlations induced by the boundary action. This is work in progress jointly with Panagiota Birmpa and Patricia Gonçalves.