Embedding theorems with an exponential weight on the real semiaxis

We state embedding theorems between spaces of functions defined on the real semi-axis, which can grow exponentially both at 0 and at +∞

Spectral comparison of compound cocycles generated by delay equations in Hilbert spaces

We study linear cocycles generated by nonautonomous delay equations in a proper Hilbert space and their extensions (compound cocycles) to exterior powers. Armed with the recently developed version of the Frequency Theorem, we develop analytic perturbation techniques for comparison of spectral properties between such cocycles and cocycles generated by stationary equations. In particular, the developed machinery is applied for studying uniform exponential dichotomies and obtaining effective dimension estimates for invariant sets arising in nonlinear problems. Our conditions are given by strict frequency inequalities involving resolvents of additive compound operators associated with stationary problems. Computing such operators requires solving a first-order PDEs with boundary conditions containing both partial derivatives and delays. However, to test frequency inequalities, the problem reduces to computation of norms of certain operators that can be done numerically and reflects computational complexity of the problem

Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models

We study the expression rates of deep neural networks (DNNs for short) for option prices written on baskets of d risky assets whose log-returns are modelled by a multivariate Lévy process with general correlation structure of jumps. We establish sufficient conditions on the characteristic triplet of the Lévy process X that ensure ε error of DNN expressed option prices with DNNs of size that grows polynomially with respect to O(ε−1), and with constants implied in O(⋅) which grow polynomially in d, thereby overcoming the curse of dimensionality (CoD) and justifying the use of DNNs in financial modelling of large baskets in markets with jumps. In addition, we exploit parabolic smoothing of Kolmogorov partial integro-differential equations for certain multivariate Lévy processes to present alternative architectures of ReLU (“rectified linear unit”) DNNs that provide ε expression error in DNN size O(|log(ε)|a) with exponent a proportional to d, but with constants implied in O(⋅) growing exponentially with respect to d. Under stronger, dimension-uniform non-degeneracy conditions on the Lévy symbol, we obtain algebraic expression rates of option prices in exponential Lévy models which are free from the curse of dimensionality. In this case, the ReLU DNN expression rates of prices depend on certain sparsity conditions on the characteristic Lévy triplet. We indicate several consequences and possible extensions of the presented results.ISSN:0949-2984ISSN:1432-112

The Quantum Field Theory of K-mouflage

We consider K-mouflage models which are K-essence theories coupled to matter. We analyse their quantum properties and in particular the quantum corrections to the classical Lagrangian. We setup the renormalisation programme for these models and show that K-mouflage theories involve a recursive construction whereby each set of counter-terms introduces new divergent quantum contributions which in turn must be subtracted by new counter-terms. This tower of counter-terms can be constructed by recursion and allows one to calculate the finite renormalised action of the model. In particular, the classical action is not renormalised and the finite corrections to the renormalised action contain only higher derivative operators. We establish an operational criterion for classicality, where the corrections to the classical action are negligible, and show that this is satisfied in cosmological and astrophysical situations for (healthy) K-mouflage models which pass the solar system tests. We also find that these models are quantum stable around astrophysical and cosmological backgrounds. We then consider the possible embedding of the K-mouflage models in an Ultra-Violet completion. We find that the healthy models which pass the solar system tests all violate the positivity constraint which would follow from the unitarity of the putative UV completion, implying that these healthy K-mouflage theories have no UV completion. We then analyse their behaviour at high energy and we find that the classicality criterion is satisfied in the vicinity of a high energy collision implying that the classical K-mouflage theory can be applied in this context. Moreover, the classical description becomes more accurate as the energy increases, in a way compatible with the classicalisation concept.Comment: 39 pages, 6 figure