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- ItemA functional limit theorem for limit order books(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Bayer, Christian; Horst, Ulrich; Qiu, JinniaoWe consider a stochastic model for the dynamics of the two-sided limit order book (LOB). For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges to a continuous-time limit when the order arrival rates tend to infinity, the impact of an individual order arrival on the book as well as the tick size tend to zero. The limits of the standing buy and sell volume densities are described by two linear stochastic partial differential equations, which are coupled with a two-dimensional reflected Brownian motion that is the limit of the best bid and ask price processes.
- ItemAdaptive behaviour in a predator-prey model leads to multiple equilibrium states(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Pimenov, Alexander; Korobeinikov, Andrei; Rachinskii, DmitriiThere is evidence that multiple stable equilibrium states are possible in real-life ecological systems. In order to verify a hypothesis that such a multitude of equilibrium states can be caused by adapting of animal behaviour to changes of environmental conditions, we consider a simple predator-prey model where prey changes a mode of behaviour in response to the pressure of predation. This model exhibits two stable coexisting equilibrium states with basins of attraction separated by a separatrix of a saddle point.
- ItemBootstrap confidence sets under a model misspecification(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Spokoiny, Vladimir; Zhilova, MayyaA multiplier bootstrap procedure for construction of likelihood-based confidence sets is considered for finite samples and possible model misspecification. Theoretical results justify the bootstrap consistency for small or moderate sample size and allow to control the impact of the parameter dimension: the bootstrap approximation works if the ratio of cube of the parameter dimension to the sample size is small. The main result about bootstrap consistency continues to apply even if the underlying parametric model is misspecified under the so called Small Modeling Bias condition. In the case when the true model deviates significantly from the considered parametric family, the bootstrap procedure is still applicable but it becomes a bit conservative: the size of the constructed confidence sets is increased by the modeling bias. We illustrate the results with numerical examples of misspecified constant and logistic regressions.
- ItemThick points for Gaussian free fields with different cut-offs(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Cipriani, Alessandra; Hazra, Rajat SubhraMassive and massless Gaussian free fields can be described as generalized Gaussian processes indexed by an appropriate space of functions. In this article we study various approaches to approximate these fields and look at the fractal properties of the thick points of their cut-offs. Under some sufficient conditions for a centered Gaussian process with logarithmic variance we study the set of thick points and derive their Hausdorff dimension. We prove that various cut-offs for Gaussian free fields satisfy these assumptions. We also give sufficient conditions for comparing thick points of different cut-offs.
- ItemOn a fractional harmonic replacement(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Dipierro, Serena; Valdinoci, EnricoGiven s e (0, 1), we consider the problem of minimizing the Gagliardo seminorm in Hs with prescribed condition outside the ball and under the further constraint of attaining zero value in a given set K. We investigate how the energy changes in dependence of such set. In particular, under mild regularity conditions, we show that adding a set A to K increases the energy of at most the measure of A (this may be seen as a perturbation result for small sets A). Also, we point out a monotonicity feature of the energy with respect to the prescribed sets and the boundary conditions.
- ItemHamiltonian framework for short optical pulses(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Amiranashvili, ShalvaPhysics of short optical pulses is an important and active research area in nonlinear optics. In what follows we theoretically consider the most extreme representatives of short pulses that contain only several oscillations of electromagnetic field. Description of such pulses is traditionally based on envelope equations and slowly varying envelope approximation, despite the fact that the envelope is not ?slow? and, moreover, there is no clear definition of such a ?fast? envelope. This happens due to another paradoxical feature: the standard (envelope) generalized nonlinear Schrödinger equation yields very good correspondence to numerical solutions of full Maxwell equations even for few-cycle pulses, a thing that should not be. In what follows we address ultrashort optical pulses using Hamiltonian framework for nonlinear waves. As it appears, the standard optical envelope equation is just a reformulation of general Hamiltonian equations. In a sense, no approximations are required, this is why the generalized nonlinear Schrödinger equation is so effective. Moreover, the Hamiltonian framework greatly contributes to our understanding of ?fast? envelope, ultrashort solitons, stability and radiation of optical pulses. Even the inclusion of dissipative terms is possible making the Hamiltonian approach an universal theoretical tool also in extreme nonlinear optics.
- ItemCorners and edges always scatter(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Elschner, Johannes; Hu, GuanghuiConsider time-harmonic acoustic scattering problems governed by the Helmholtz equation in two and three dimensions. We prove that bounded penetrable obstacles with corners or edges scatter every incident wave nontrivially, provided the function of refractive index is real-analytic. Moreover, if such a penetrable obstacle is a convex polyhedron or polygon, then its shape can be uniquely determined by the far-field pattern over all observation directions incited by a single incident wave. Our arguments are elementary and rely on the expansion of solutions to the Helmholtz equation.
- ItemExistence of bounded discrete steady state solutions of the van Roosbroeck system with monotone Fermi-Dirac statistic functions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Gärtner, KlausIf the statistic function is modified, the equations can be derived by a variational formulation or just using a generalized Einstein relation. In both cases a dissipative generalization of the Scharfetter-Gummel scheme citeSch_Gu, understood as a one-dimensional constant current approximation, is derived for strictly monotone coefficient functions in the elliptic operator $nabla cdot bal ff(v) nabla $, $v$ chemical potential, while the hole density is defined by $p=cal F(v)le e^v.$ A closed form integration of the governing equation would simplify the practical use, but mean value theorem based results are sufficient to prove existence of bounded discrete steady state solutions on any boundary conforming Delaunay grid. These results hold for any piecewise, continuous, and monotone approximation of $bal ff(v)$ and $cal F(v)$.
- ItemOn thermodynamical couplings of quantum mechanics and macroscopic systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Mielke, AlexanderPure quantum mechanics can be formulated as a Hamiltonian system in terms of the Liouville equation for the density matrix. Dissipative effects are modeled via coupling to a macroscopic system, where the coupling operators act via commutators. Following Öttinger (2010) we use the GENERIC framework to construct thermodynamically consistent evolution equations as a sum of a Hamiltonian and a gradient-flow contribution, which satisfy a particular non-interaction condition: q̇ = J(q)DE(q) + K(q)DS(q). We give three applications of the theory. First, we consider a finite-dimensional quantum system that is coupled to a finite number of simple heat baths, each of which is described by a scalar temperature variable. Second, we model quantum system given by a one-dimensional Schrödinger operator connected to a onedimensional heat equation on the left and on the right. Finally, we consider thermoopto-electronics, where the Maxwell-Bloch system of optics is coupled to the energydrift-diffusion system for semiconductor electronics.
- ItemDeriving amplitude equations via evolutionary [Gamma]-convergence(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Mielke, AlexanderWe discuss the justification of the GinzburgLandau equation with real coefficients as an amplitude equation for the weakly unstable one-dimensional SwiftHohenberg equation. In contrast to classical justification approaches we employ the method of evolutionary [Gamma]-convergence by reformulating both equation as gradient systems. Using a suitable linear transformation we show [Gamma]-convergence of the associated energies in suitable function spaces. The limit passage of the time-dependent problem relies on the recent theory of evolutionary variational inequalities for families of uniformly convex functionals as developed by Daneri and Savare 2010. In the case of a cubic energy it suffices that the initial conditions converge strongly in L2, while for the case of a quadratic nonlinearity we need to impose weak convergence in H1. However, we do not need wellpreparedness of the initial conditions.