Beyond conventional Runge–Kutta methods in numerical integration of ODEs and DAEs by use of structures and local models

There are two parts in this paper. In the first part we consider an overdetermined system of differential-algebraic equations (DAEs). We are particularly concerned with Hamiltonian and Lagrangian systems with holonomic constraints. The main motivation is in finding methods based on Gauss coefficients, preserving not only the constraints, symmetry, symplecticness, and variational nature of trajectories of holonomically constrained Hamiltonian and Lagrangian systems, but also having optimal order of convergence. The new class of (s,s)$(s,s)$-Gauss–Lobatto specialized partitioned additive Runge–Kutta (SPARK) methods uses greatly the structure of the DAEs and possesses all desired properties. In the second part we propose a unified approach for the solution of ordinary differential equations (ODEs) mixing analytical solutions and numerical approximations. The basic idea is to consider local models which can be solved efficiently, for example analytically, and to incorporate their solution into a global procedure based on standard numerical integration methods for the correction. In order to preserve also symmetry we define the new class of symmetrized Runge–Kutta methods with local model (SRKLM).     

A Gutzwiller type formula for a reduced Hamiltonian within the framework of symmetry

For a classical Hamiltonian with a finite group of symmetries, we give semi-classical asymptotics in a neighbourhood of an energy E of a regularized spectral density of the quantum Hamiltonian restricted to symmetry subspaces of Peter–Weyl defined by irreducible characters of the group. If we suppose that the energy level ${\Sigma }_E$ is compact, non-critical, and that its periodic orbits are non-degenerate, we get a Gutzwiller type formula for the reduced Hamiltonian, whose oscillating part involves the symmetry properties of closed trajectories of ${\Sigma }_E$. To cite this article: R. Cassanas, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

q-Symmetry in Condensed Matter Physics: an Effective Generalized Hubbard Model with Quantum Symmetry

Using the generalized LangFirsov transformation, we reduce the Hubbard model with phonons (which, under particular circumstances, has recently been shown to exhibit the quantum symmetry su(2)[su(2)]_q into an effective electron Hamiltonian. The effective Hamiltonian is explicitly proved to maintain the quantum symmetry, which survives even when relaxing one of the constraints on the Hamiltonian parameters that must hold in the model with phonons. We investigate under which circumstances some superconducting eigenstates of the effective Hamiltonian, which can be built thanks to the quantum symmetry, could become metastable. Bibliography: 12 titles.     

Subgeometric rates of convergence of f-ergodic strong Markov processes

We provide a condition in terms of a supermartingale property for a functional of the Markov process, which implies (a) f$f$-ergodicity of strong Markov processes at a subgeometric rate, and (b) a moderate deviation principle for an integral (bounded) functional. An equivalent condition in terms of a drift inequality on the extended generator is also given. Results related to (f,r)$(f,r)$-regularity of the process, of some skeleton chains and of the resolvent chain are also derived. Applications to specific processes are considered, including elliptic stochastic differential equations, Langevin diffusions, hypoelliptic stochastic damping Hamiltonian systems and storage models.     

Nonequilibrium Statistical Mechanics of Weakly Stochastically Perturbed System of Oscillators

We consider a finite region of a d-dimensional lattice, \({d \in \mathbb{N}}\), of weakly coupled harmonic oscillators. The coupling is provided by a nearest-neighbour potential (harmonic or not) of size \({\varepsilon}\). Each oscillator weakly interacts by force of order \({\varepsilon}\) with its own stochastic Langevin thermostat of arbitrary positive temperature. We investigate limiting as \({\varepsilon \rightarrow 0}\) behaviour of solutions of the system and of the local energy of oscillators on long-time intervals of order \({\varepsilon^{-1}}\) and in a stationary regime. We show that it is governed by an effective equation which is a dissipative SDE with nondegenerate diffusion. Next, we assume that the interaction potential is of size \({\varepsilon \lambda}\), where \({\lambda}\) is another small parameter, independent from \({\varepsilon}\). Solutions corresponding to this scaling describe small low temperature oscillations. We prove that in a stationary regime, under the limit \({\varepsilon \rightarrow 0}\), the main order in \({\lambda}\) of the averaged Hamiltonian energy flow is proportional to the gradient of temperature. We show that the coefficient of proportionality, which we call the conductivity, admits a representation through stationary space–time correlations of the energy flow. Most of the results and convergences we obtain are uniform with respect to the number of oscillators in the system.     

A plethora of three-dimensional periodic travelling gravity-capillary water waves with multipulse transverse profiles

This article presents a rigorous existence theory for three-dimensional gravity-capillary water waves which are uniformly translating and periodic in one spatial direction x and have the profile of a uni- or multipulse solitary wave in the other z. The waves are detected using a combination of Hamiltonian spatial dynamics and homoclinic Lyapunov-Schmidt theory. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which z is the timelike variable, and a family of points P_k,k+1, k = 1,2,... in its two-dimensional parameter space is identified at which a Hamiltonian 0²0² resonance takes place (the zero eigenspace and generalised eigenspace are respectively two and four dimensional). The point P_k,k+1 is precisely that at which a pair of two-dimensional periodic linear travelling waves with frequency ratio k:k+1 simultaneously exist (Wilton ripples). A reduction principle is applied to demonstrate that the problem is locally equivalent to a four-dimensional Hamiltonian system near P_k,k+1. It is shown that a Hamiltonian real semisimple 1:1 resonance, where two geometrically double real eigenvalues exist, arises along a critical curve R_k,k+1 emanating from P_k,k+1. Unipulse transverse homoclinic solutions to the reduced Hamiltonian system at points of R_k,k+1 near P_k,k+1 are found by a scaling and perturbation argument, and the homoclinic Lyapunov-Schmidt method is applied to construct an infinite family of multipulse homoclinic solutions which resemble multiple copies of the unipulse solutions.     

Black–Scholes in a CEV random environment

Classical (Itô diffusions) stochastic volatility models are not able to capture the steepness of small-maturity implied volatility smiles. Jumps, in particular exponential Lévy and affine models, which exhibit small-maturity exploding smiles, have historically been proposed to remedy this (see Tankov in Pricing and hedging in exponential Lévy models: review of recent results. Paris-Princeton Lecture Notes in Mathematical Finance, Springer, Berlin, 2010 for an overview), and more recently rough volatility models (Alòs et al. in On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility. Finance Stoch 11(4):571–589, 2007, Fukasawa in Asymptotic analysis for stochastic volatility: martingale expansion. Finance Stoch 15:635–654, 2011). We suggest here a different route, randomising the Black–Scholes variance by a CEV-generated distribution, which allows us to modulate the rate of explosion (through the CEV exponent) of the implied volatility for small maturities. The range of rates includes behaviours similar to exponential Lévy models and fractional stochastic volatility models.     

Random perturbations of 1:1-resonant systems with SO(2) symmetry

^** Email: narayananramakrishnan{at}maxtor.com^*** Email: navam{at}uiuc.edu The near-resonant motion of trajectories of an integrable systemsubject to small dissipation and random perturbations is analysed.Under an appropriate change of time, we identify a reduced model.Our principal technique of dimensional reduction will be themethod of stochastic averaging for non-linear systems with smallnoise. A scheme to obtain the averaged motion of trajectoriesnear resonances is developed. The method of reduction is presentedusing the example of a two-degree-of-freedom Hamiltonian systemwith SO(2) symmetry subject to random perturbations. The averagedmotion of trajectories close to the 1:1-resonance surface isobtained and the effects of random perturbations on the near-resonantdynamics are analysed.     

Sufficient Stochastic Maximum Principle for Discounted Control Problem

In this article, the sufficient Pontryagin’s maximum principle for infinite horizon discounted stochastic control problem is established. The sufficiency is ensured by an additional assumption of concavity of the Hamiltonian function. Throughout the paper, it is assumed that the control domain \(U\) is a convex bounded set and the control may enter the diffusion term of the state equation. The general results are applied to the controlled stochastic logistic equation of population dynamics.     

Symmetries of periodic solutions for planar potential systems

In this article we discuss the symmetries of periodic solutions to Hamiltonian systems with two degrees of freedom in mechanical form. The possible symmetries of such periodic trajectories are generated by spatial symmetries (a finite subgroup of , phase-shift symmetries (the circle group , and a time-reversing symmetry (associated with mechanical form). We focus on the symmetries and structures of the trajectories in configuration space (), showing that special properties such as self-intersections and brake orbits are consequences of symmetry.

     

Models of stochastic geometry — A survey

This paper discusses some models of stochastic geometry which are of potential interest for operations research. These are the Boolean model, a certain model for random compact sets and marked point processes. The Boolean model is a generalization of the well-known queueing systemM/G/. The random compact set model may be useful for modelling spatial spreading processes such as fires, cancers or holes in the Earth's surface. Marked point processes are used here as models of forests and used for a statistical study of the spatial distribution of damaged trees.Extended version of an Invited Lecture on the 16th Symposium for OR in Hamburg 1992.     

Stochastic Evolution of Augmented Born–Infeld Equations

This paper compares the results of applying a recently developed method of stochastic uncertainty quantification designed for fluid dynamics to the Born–Infeld model of nonlinear electromagnetism. The similarities in the results are striking. Namely, the introduction of Stratonovich cylindrical noise into each of their Hamiltonian formulations introduces stochastic Lie transport into their dynamics in the same form for both theories. Moreover, the resulting stochastic partial differential equations retain their unperturbed form, except for an additional term representing induced Lie transport by the set of divergence-free vector fields associated with the spatial correlations of the cylindrical noise. The explanation for this remarkable similarity lies in the method of construction of the Hamiltonian for the Stratonovich stochastic contribution to the motion in both cases, which is done via pairing spatial correlation eigenvectors for cylindrical noise with the momentum map for the deterministic motion. This momentum map is responsible for the well-known analogy between hydrodynamics and electromagnetism. The momentum map for the Maxwell and Born–Infeld theories of electromagnetism treated here is the 1-form density known as the Poynting vector. Two appendices treat the Hamiltonian structures underlying these results.

     

Group-Theoretical Analysis of Variable Coefficient Nonlinear Telegraph Equations

Given a class \(\mathcal{F(\theta)}\) of differential equations with arbitrary element θ, the problems of symmetry group, nonclassical symmetry and conservation law classifications are to determine for each member \(f\in\mathcal{F(\theta)}\) the structure of its Lie symmetry group G _f , conditional symmetry Q _f and conservation law \(\mathop {\rm CL}\nolimits _{f}\) under some proper equivalence transformations groups.In this paper, an extensive investigation of these three aspects is carried out for the class of variable coefficient (1+1)-dimensional nonlinear telegraph equations with coefficients depending on the space variable f(x)u _tt =(g(x)H(u)u _x ) _x +h(x)K(u)u _x . The usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements are first constructed. Then using the technique of variable gauges of arbitrary elements under equivalence transformations, we restrict ourselves to the symmetry group classifications for the equations with two different gauges g=1 and g=h. In order to get the ultimate classification, the method of furcate split is also used and consequently a number of new interesting nonlinear invariant models which have non-trivial invariance algebra are obtained. As an application, exact solutions for some equations which are singled out from the classification results are constructed by the classical method of Lie reduction.The classification of nonclassical symmetries for the classes of differential equations with gauge g=1 is discussed within the framework of singular reduction operator. This enabled to obtain some exact solutions of the nonlinear telegraph equation which are invariant under certain conditional symmetries.Using the direct method, we also carry out two classifications of local conservation laws up to equivalence relations generated by both usual and extended equivalence groups. Equivalence with respect to these groups and correct choice of gauge coefficients of equations play the major role for simple and clear formulation of the final results.     

Continuous Local Time of a Purely Atomic Immigration Superprocess with Dependent Spatial Motion

Abstract

A purely atomic immigration superprocess with dependent spatial motion in the space of tempered measures is constructed as the unique strong solution of a stochastic integral equation driven by Poisson processes based on the excursion law of a Feller branching diffusion, which generalizes the work of Dawson and Li [3 Dawson , D.A. , and Li , Z.H. 2003 . Construction of immigration superprocesses with dependent spatial motion from one-dimensional excursions . Probability Theory and Related Fields 127 : 37 – 61 . [Google Scholar]]. As an application of the stochastic equation, it is proved that the superprocess possesses a local time which is Hölder continuous of order α for every α < 1/2. We establish two scaling limit theorems for the immigration superprocess, from which we derive scaling limits for the corresponding local time.     

Amalgamating players,symmetry, and the Banzhaf value

We suggest new characterizations of the Banzhaf value without the symmetry axiom, which reveal that the characterizations by Lehrer (Int J Game Theory 17:89–99, 1988) and Nowak (Int J Game Theory 26:137–141, 1997) as well as most of the characterizations by Casajus (Theory Decis 71:365–372, 2011b) are redundant. Further, we explore symmetry implications of Lehrer’s 2-efficiency axiom.     

Arithmetic Asian Options under Stochastic Delay Models

Abstract

Motivated by the increasing interest in past-dependent asset pricing models, shown in recent years by market practitioners and prominent authors such as Hobson and Rogers (1998 Hobson, D. and Rogers, L. C. G. 1998. Complete models with stochastic volatility. Mathematical Finance, 8(1): 27–48.  [Google Scholar], Complete models with stochastic volatility, Mathematical Finance, 8(1), pp. 27–48), we explore option pricing techniques for arithmetic Asian options under a stochastic delay differential equation approach. We obtain explicit closed-form expressions for a number of lower and upper bounds and compare their accuracy numerically.