Refine
Year of publication
- 2005 (5) (remove)
Language
- English (5) (remove)
Project
- E2 (3)
Application Area
- E (3)
The background for the general mathematical link between utility and information
theory investigated in this paper is a simple financial market model with two kinds of small
traders: less informed traders and insiders whose extra information is represented by an
enlargement of the other agents' filtration. The expected logarithmic utility increment,
i.e. the difference of the insider's and the less informed trader's expected logarithmic
utility is described in terms of the information drift, i.e. the drift one has to eliminate
in order to perceive the price dynamics as a martingale from the insider's perspective.
On the one hand, we describe the information drift in a very general setting by natural
quantities expressing the probabilistic better informed view of the world. This on the
other hand allows us to identify the additional utility by entropy related quantities known
from information theory. In particular in a complete market in which the insider has some
fixed additional information during the entire trading interval, its utility increment can
be represented by the Shannon information of his extra knowledge. For general markets,
and in some particular examples, we provide estimates of maximal utility by information
inequalities.
Let (Gt) be an enlargement of the filtration (Ft). Jeulin and Jacod
discussed a sufficient criterion for the inheritance of the semimartingale property
when passing to the larger filtration. We provide alternative proofs of their results
in a more general setting by using decoupling measures and Girsanov's changes of
measure. We derive necessary and sufficient conditions for the embedding of vector
spaces of (Ft)-semimartingales into spaces of (Gt)-semimartingales to be continuous
in terms of generalized entropies of the information increment.
The subject of the present paper is a simplified model for a symmetric bistable system with memory
or delay, the reference model, which in the presence of noise exhibits a phenomenon similar to what
is known as stochastic resonance. The reference model is given by a one dimensional parametrized
stochastic differential equation with point delay, basic properties whereof we check.
With a view to capturing the effective dynamics and, in particular, the resonance-like behavior of
the reference model we construct a simplified or reduced model, the two state model, first in discrete
time, then in the limit of discrete time tending to continuous time. The main advantage of the
reduced model is that it enables us to explicitly calculate the distribution of residence times which
in turn can be used to characterize the phenomenon of noise-induced resonance.
Drawing on what has been proposed in the physics literature, we outline a heuristic method for
establishing the link between the two state model and the reference model. The resonance characteristics
developed for the reduced model can thus be applied to the original model.
We consider potential type dynamical systems in finite dimensions with two meta-stable states.
They are subject to two sources of perturbation: a slow external periodic perturbation of period T
and a small Gaussian random perturbation of intensity ", and therefore mathematically described as
weakly time inhomogeneous diffusion processes. A system is in stochastic resonance provided the small
noisy perturbation is tuned in such a way that its random trajectories follow the exterior periodic
motion in an optimal fashion, i.e. for some optimal intensity "(T). The physicists' favorite measures
of quality of periodic tuning -- and thus stochastic resonance -- such as spectral power amplification or
signal-to-noise ratio have proven to be defective. They are not robust w.r.t. effective model reduction,
i.e. for the passage to a simplified finite state Markov chain model reducing the dynamics to a pure
jumping between the meta-stable states of the original system. An entirely probabilistic notion of
stochastic resonance based on the transition dynamics between the domains of attraction of the meta-
stable states -- and thus failing to suffer from this robustness defect -- was proposed before in the
context of one-dimensional discusions. It is investigated for higher dimensional systems here, by using
extensions and refinements of the Freidlin-Wentzell theory of large deviations for time homogeneous
diffusions. Large deviation principles developed for weakly time inhomogeneous diffusions prove to be
key tools for a treatment of the problem of diffusion exit from a domain and thus for the approach of
stochastic resonance via transition probabilities between meta-stable sets.