1,107 research outputs found
Equilibrium Asset Pricing with Transaction Costs
We study risk-sharing economies where heterogenous agents trade subject to
quadratic transaction costs. The corresponding equilibrium asset prices and
trading strategies are characterised by a system of nonlinear, fully-coupled
forward-backward stochastic differential equations. We show that a unique
solution generally exists provided that the agents' preferences are
sufficiently similar. In a benchmark specification with linear state dynamics,
the illiquidity discounts and liquidity premia observed empirically correspond
to a positive relationship between transaction costs and volatility.Comment: 32 pages, forthcoming in 'Finance and Stochastics
A Hybrid Algorithm Based on Optimal Quadratic Spline Collocation and Parareal Deferred Correction for Parabolic PDEs
Parareal is a kind of time parallel numerical methods for time-dependent systems. In this paper, we consider a general linear parabolic PDE, use optimal quadratic spline collocation (QSC) method for the space discretization, and proceed with the parareal technique on the time domain. Meanwhile, deferred correction technique is also used to improve the accuracy during the iterations. In fact, the optimal QSC method is a correction of general QSC method. Along the temporal direction we embed the iterations of deferred correction into parareal to construct a hybrid method, parareal deferred correction (PDC) method. The error estimation is presented and the stability is analyzed. To save computational cost, we find out a simple way to balance the two kinds of iterations as much as possible. We also argue that the hybrid algorithm has better system efficiency and costs less running time. Numerical experiments by multicore computers are attached to exhibit the effectiveness of the hybrid algorithm
Generalized Green-Kubo formulas for fluids with impulsive, dissipative, stochastic and conservative interactions
We present a generalization of the Green-Kubo expressions for thermal
transport coefficients in complex fluids of the generic form, , i.e.
a sum of an instantaneous transport coefficient , and a time
integral over a time correlation function in a state of thermal equilibrium
between a current and a transformed current . The streaming
operator generates the trajectory of a dynamical variable
when used inside the thermal average . These
formulas are valid for conservative, impulsive (hard spheres), stochastic and
dissipative forces (Langevin fluids), provided the system approaches a thermal
equilibrium state. In general and ,
except for the case of conservative forces, where the equality signs apply. The
most important application in the present paper is the hard sphere fluid.Comment: 14 pages, no figures. Version 2: expanded Introduction and section II
specifying the classes of fluids covered by this theory. Some references
added and typos correcte
Maximum-likelihood estimation for diffusion processes via closed-form density expansions
This paper proposes a widely applicable method of approximate
maximum-likelihood estimation for multivariate diffusion process from
discretely sampled data. A closed-form asymptotic expansion for transition
density is proposed and accompanied by an algorithm containing only basic and
explicit calculations for delivering any arbitrary order of the expansion. The
likelihood function is thus approximated explicitly and employed in statistical
estimation. The performance of our method is demonstrated by Monte Carlo
simulations from implementing several examples, which represent a wide range of
commonly used diffusion models. The convergence related to the expansion and
the estimation method are theoretically justified using the theory of Watanabe
[Ann. Probab. 15 (1987) 1-39] and Yoshida [J. Japan Statist. Soc. 22 (1992)
139-159] on analysis of the generalized random variables under some standard
sufficient conditions.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1118 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A hybrid parareal spectral deferred corrections method
The parareal algorithm introduced in 2001 by Lions, Maday, and Turinici is an iterative method for the parallelization of the numerical solution of ordinary differential equations or partial differential equations discretized in the temporal direction. The temporal interval of interest is partitioned into successive domains which are assigned to separate processor units. Each iteration of the parareal algorithm consists of a high accuracy solution procedure performed in parallel on each domain using approximate initial conditions and a serial step which propagates a correction to the initial conditions through the entire time interval. The original method is designed to use classical single-step numerical methods for both of these steps. This paper investigates a variant of the parareal algorithm first outlined by Minion and Williams in 2008 that utilizes a deferred correction strategy within the parareal iterations. Here, the connections between parareal, parallel deferred corrections, and a hybrid parareal-spectral deferred correction method are further explored. The parallel speedup and efficiency of the hybrid methods are analyzed, and numerical results for ODEs and discretized PDEs are presented to demonstrate the performance of the hybrid approach
On the stability of robust dynamical low-rank approximations for hyperbolic problems
The dynamical low-rank approximation (DLRA) is used to treat high-dimensional problems that arise in such diverse fields as kinetic transport and uncertainty quantification. Even though it is well known that certain spatial and temporal discretizations when combined with the DLRA approach can result in numerical instability, this phenomenon is poorly understood. In this paper we perform a L2 stability analysis for the corresponding nonlinear equations of motion. This reveals the source of the instability for the projector splitting integrator when first discretizing the equations and then applying the DLRA. Based on this we propose a projector splitting integrator, based on applying DLRA to the continuous system before performing the discretization, that recovers the classic CFL condition. We also show that the unconventional integrator has more favorable stability properties and explain why the projector splitting integrator performs better when approximating higher moments, while the unconventional integrator is generally superior for first order moments. Furthermore, an efficient and stable dynamical low-rank update for the scattering term in kinetic transport is proposed. Numerical experiments for kinetic transport and uncertainty quantification, which confirm the results of the stability analysis, are presented
Differential Equation Models in Applied Mathematics
The present book contains the articles published in the Special Issue “Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges” of the MDPI journal Mathematics. The Special Issue aimed to highlight old and new challenges in the formulation, solution, understanding, and interpretation of models of differential equations (DEs) in different real world applications. The technical topics covered in the seven articles published in this book include: asymptotic properties of high order nonlinear DEs, analysis of backward bifurcation, and stability analysis of fractional-order differential systems. Models oriented to real applications consider the chemotactic between cell species, the mechanism of on-off intermittency in food chain models, and the occurrence of hysteresis in marketing. Numerical aspects deal with the preservation of mass and positivity and the efficient solution of Boundary Value Problems (BVPs) for optimal control problems. I hope that this collection will be useful for those working in the area of modelling real-word applications through differential equations and those who care about an accurate numerical approximation of their solutions. The reading is also addressed to those willing to become familiar with differential equations which, due to their predictive abilities, represent the main mathematical tool for applying scenario analysis to our changing world
Statistics of Avalanches with Relaxation, and Barkhausen Noise: A Solvable Model
We study a generalization of the Alessandro-Beatrice-Bertotti-Montorsi (ABBM)
model of a particle in a Brownian force landscape, including retardation
effects. We show that under monotonous driving the particle moves forward at
all times, as it does in absence of retardation (Middleton's theorem). This
remarkable property allows us to develop an analytical treatment. The model
with an exponentially decaying memory kernel is realized in Barkhausen
experiments with eddy-current relaxation, and has previously been shown
numerically to account for the experimentally observed asymmetry of
Barkhausen-pulse shapes. We elucidate another qualitatively new feature: the
breakup of each avalanche of the standard ABBM model into a cluster of
sub-avalanches, sharply delimited for slow relaxation under quasi-static
driving. These conditions are typical for earthquake dynamics. With relaxation
and aftershock clustering, the present model includes important ingredients for
an effective description of earthquakes. We analyze quantitatively the limits
of slow and fast relaxation for stationary driving with velocity v>0. The
v-dependent power-law exponent for small velocities, and the critical driving
velocity at which the particle velocity never vanishes, are modified. We also
analyze non-stationary avalanches following a step in the driving magnetic
field. Analytically, we obtain the mean avalanche shape at fixed size, the
duration distribution of the first sub-avalanche, and the time dependence of
the mean velocity. We propose to study these observables in experiments,
allowing to directly measure the shape of the memory kernel, and to trace eddy
current relaxation in Barkhausen noise.Comment: 39 pages, 26 figure
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