5,382 research outputs found
Uniform global attractors for the nonautonomous 3D Navier-Stokes equations
We obtain the existence and the structure of the weak uniform (with respect
to the initial time) global attractor and construct a trajectory attractor for
the 3D Navier-Stokes equations (NSE) with a fixed time-dependent force
satisfying a translation boundedness condition. Moreover, we show that if the
force is normal and every complete bounded solution is strongly continuous,
then the uniform global attractor is strong, strongly compact, and solutions
converge strongly toward the trajectory attractor. Our method is based on
taking a closure of the autonomous evolutionary system without uniqueness,
whose trajectories are solutions to the nonautonomous 3D NSE. The established
framework is general and can also be applied to other nonautonomous dissipative
partial differential equations for which the uniqueness of solutions might not
hold. It is not known whether previous frameworks can also be applied in such
cases as we indicate in open problems related to the question of uniqueness of
the Leray-Hopf weak solutions.Comment: 28 pages. arXiv admin note: text overlap with arXiv:math/060935
On the super replication price of unbounded claims
In an incomplete market the price of a claim f in general cannot be uniquely
identified by no arbitrage arguments. However, the ``classical'' super
replication price is a sensible indicator of the (maximum selling) value of the
claim. When f satisfies certain pointwise conditions (e.g., f is bounded from
below), the super replication price is equal to sup_QE_Q[f], where Q varies on
the whole set of pricing measures. Unfortunately, this price is often too high:
a typical situation is here discussed in the examples. We thus define the less
expensive weak super replication price and we relax the requirements on f by
asking just for ``enough'' integrability conditions. By building up a proper
duality theory, we show its economic meaning and its relation with the
investor's preferences. Indeed, it turns out that the weak super replication
price of f coincides with sup_{Q\in M_{\Phi}}E_Q[f], where M_{\Phi} is the
class of pricing measures with finite generalized entropy (i.e., E[\Phi
(\frac{dQ}{dP})]<\infty) and where \Phi is the convex conjugate of the utility
function of the investor.Comment: Published at http://dx.doi.org/10.1214/105051604000000459 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Trajectory attractors for the Sun-Liu model for nematic liquid crystals in 3D
In this paper we prove the existence of a trajectory attractor (in the sense
of V.V. Chepyzhov and M.I. Vishik) for a nonlinear PDE system coming from a 3D
liquid crystal model accounting for stretching effects. The system couples a
nonlinear evolution equation for the director d (introduced in order to
describe the preferred orientation of the molecules) with an incompressible
Navier-Stokes equation for the evolution of the velocity field u. The technique
is based on the introduction of a suitable trajectory space and of a metric
accounting for the double-well type nonlinearity contained in the director
equation. Finally, a dissipative estimate is obtained by using a proper
integrated energy inequality. Both the cases of (homogeneous) Neumann and
(non-homogeneous) Dirichlet boundary conditions for d are considered.Comment: 32 page
Congestion Management in European Power Networks: Criteria to Assess the Available Options
EU Member States are pursuing large scale investment in renewable generation in order to meet a 2020 target to source 20% of total energy sources by renewables. As the location for this new generation differs from the location of existing generation sources, and is often on the extremities of the electricity network, it will create new flow patterns and transmission needs. While congestion exists between European countries, increasing the penetration of variable sources of energy will change the current cross-border congestion profile. It becomes increasingly important for the power market design to foster the full use of existing transmission capacity and allow for robust operation even in the presence of system congestion. After identifying five criteria that an effective congestion management scheme for European countries will need, this paper critically assess to what extent the various approaches satisfy the requirements.Power market design, integrating renewables, congestion management
Arbitrage and Control Problems in Finance. Presentation.
The theory of asset pricing takes its roots in the Arrow-Debreu model (see,for instance, Debreu 1959, Chap. 7), the Black and Scholes (1973) formula,and the Cox and Ross (1976) linear pricing model. This theory and its link to arbitrage has been formalized in a general framework by Harrison and Kreps (1979), Harrison and Pliska (1981, 1983), and Du¹e and Huang (1986). In these models, security markets are assumed to be frictionless: securities can be sold short in unlimited amounts, the borrowing and lending rates are equal, and there is no transaction cost. The main result is that the price process of traded securities is arbitrage free if and only if there exists some equivalent probability measure that transforms it into a martingale, when normalized by the numeraire. Contingent claims can then be priced by taking the expected value of their (normalized) payo§ with respect to any equivalent martingale measure. If this value is unique, the claim is said to be priced by arbitrage and it can be perfectly hedged (i.e. duplicated) by dynamic trading. When the markets are dynamically complete, there is only one such a and any contingent claim is priced by arbitrage. The of each state of the world for this probability measure can be interpreted as the state price of the economy (the prices of $1 tomorrow in that state of the world) as well as the marginal utilities (for consumption in that state of the world) of rational agents maximizing their expected utility.arbitrage, control problem
- âŠ