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We study the effect of investor inertia on stock price fluctuations with a market microstructure
model comprising many small investors who are inactive most of the time.
It turns out that semi-Markov processes are tailor made for modelling inert investors.
With a suitable scaling, we show that when the price is driven by the market imbalance,
the log price process is approximated by a process with long range dependence
and non-Gaussian returns distributions, driven by a fractional Brownian motion. Consequently,
investor inertia may lead to arbitrage opportunities for sophisticated market
participants. The mathematical contributions are a functional central limit theorem for
stationary semi-Markov processes, and approximation results for stochastic integrals
of continuous semimartingales with respect to fractional Brownian motion.
We consider a financial market model with a large number of interacting
agents. Investors are heterogeneous in their expectations
about the future evolution of an asset price process. Their current
expectation is based on the previous states of their “neighbors” and
on a random signal about the “mood of the market.” We analyze the
asymptotics of both aggregate behavior and asset prices. We give sufficient
conditions for the distribution of equilibrium prices to converge to
a unique equilibrium, and provide a microeconomic foundation for the
use of diffusion models in the analysis of financial price fluctuations.
We consider general economies in which rational agents interact locally. The local aspect
of the interactions is designed to represent in a simple abstract way social interactions, that
is, socioeconomic environments in which markets do not mediate all of agents' choices, and
each agent's choice might be in part determined, for instance, by family, peer group, or ethnic
group effects. We study static as well as dynamic infinite horizon economies; we allow for
economies with incomplete information, and we consider jointly global and local interactions,
to integrate e.g., global externalities and markets with peer and group effects. We provide
conditions under which such economies have rational expectations equilibria.
We illustrate the effects of local interactions when agents are rational by studying in detail
the equilibrium properties of a simple economy with quadratic preferences which captures, in
turn, local preferences for conformity, habit persistence, and preferences for status or adherence
to aggregate norms of behavior.
We give sufficient conditions for a non-zero sum discounted stochastic game with
compact and convex action spaces and with norm-continuous transition probabilities,
but with possibly unbounded state space, to have a Nash equilibrium in homogeneous
Markov strategies that depends in a Lipschitz continuous manner on the current state. If
the underlying state space is compact this yields the existence of a stationary equilibrium.
Stochastic games with weakly interacting players provide a probabilistic framework within
which to study strategic behavior in models of non-market interactions.
This paper addresses the regularization of pointwise state constraints in optimal
control problems. By analyzing the associated dual problem, it is shown that the regularized problems
admit Lagrange multipliers in L2-spaces. Under a certain boundedness assumption, the solution of
the regularized problem converges to the one of the original state constrained problem. The results
of our analysis are confirmed by numerical tests.
An optimal control problem for a 2-d elliptic equation is investigated with pointwise control constraints.
This paper is concerned with discretization of the control by piecewise constant functions. The state and
the adjoint state are discretized by linear finite elements. Approximations of the optimal solution of the continuous
optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these
approximations have convergence order h2.
We provide a number of new construction techniques for cubical complexes and cubical
polytopes, and thus for cubifications (hexahedral mesh generation). As an application we
obtain an instance of a cubical 4-polytope that has a non-orientable dual manifold (a Klein
bottle). This confirms an existence conjecture of Hetyei (1995).
More systematically, we prove that every normal crossing codimension one immersion of
a compact 2-manifold into R3 is PL-equivalent to a dual manifold immersion of a cubical
4-polytope. As an instance we obtain a cubical 4-polytope with a cubation of Boy's surface
as a dual manifold immersion, and with an odd number of facets. Our explicit example has
17 718 vertices and 16 533 facets. Thus we get a parity changing operation for 3-dimensional
cubical complexes (hexa meshes); this solves problems of Eppstein, Thurston, and others.
We discuss solvers for Sylvester, Lyapunov, and Stein equations
that are available in the SLICOT Library (Subroutine
Library In COntrol Theory). These solvers offer improved
efficiency, reliability, and functionality compared to corresponding
solvers in other computer-aided control system design
packages. The performance of the SLICOT solvers is
compared with the corresponding MATLAB solvers.
The paper presents a unified approach to local likelihood estimation
for a broad class of nonparametric models, including e.g. the regression,
density, Poisson and binary response model. The method extends
the adaptive weights smoothing (AWS) procedure introduced in Polzehl
and Spokoiny (2000) in context of image denoising. Performance of the
proposed procedure is illustrated by a number of numerical examples
and applications to density or volatility estimation, classification and
estimation of the tail index parameter. We also establish a number of
important theoretical results on properties of the proposed procedure.
The adaptive weights smoothing (AWS) procedure was introduced in
Polzehl and Spokoiny (2000) in the context of image denoising. The
procedure has some remarkable properties like preservation of edges and
contrast, and (in some sense) optimal reduction of noise. The procedure
is fully adaptive and dimension free. Simulations with artificial images
show that AWS is superior to classical smoothing techniques especially
when the underlying image function is discontinuous and can be well
approximated by a piecewise constant function. However, the latter as-
sumption can be rather restrictive for a number of potential applications.
Here the AWS method is generalized to the case of an arbitrary local lin-
ear parametric structure. We also establish some important results about
properties of the AWS procedure including the so called "propagation
condition" and spatial adaptivity. The performance of the procedure is
illustrated by examples for local polynomial regression in univariate and
bivariate situations.