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Invariant measures of dynamical systems generated e. g. by difference equations
can be computed by discretizing the originally continuum state space, and
replacing the action of the generator by the transition mechanism of a Markov
chain. In fact they are approximated by stationary vectors of these Markov
chains. Here we extend this well known approximation result and the underlying
algorithm to the setting of random dynamical systems, i.e. dynamical systems
on the skew product of a probability space carrying the underlying stationary
stochasticity and the state space, a particular non-autonomous framework. The
systems are generated by difference equations driven by stationary random processes
modelled on a metric dynamical system. The approximation algorithm
involves spatial discretizations and the definition of appropriate random Markov
chains with stationary vectors converging to the random invariant measure of
the system.
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 behaviour 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 characteristic developed for the reduced model can thus be applied to the original model.
Short term climate events such as the sea surface temperature anomaly known as El Nino are financial risk sources leading to incomplete markets. To make such risk tradable, we use a market model in which a climate index provides an extra investment opinion. Given one possible market price of risk each agent can maximize the exponential utility from three sources of income: capital market, additional security, and individual risk exposure. Under an equilibrium condition the market price of risk is uniquely determined by a backward stochastic differential equation. We translate these stochastic equations into semi-linear partial differential equations for the simulation of which numerical schemes are available. We choose two simple models for sea surface temperature, and with ENSO risk exposed fisher and farmer and a nonh-exposed bank three toy agents. By simulating their optimal investment into the climat index we obtain first insight into the dynamics of the market.
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 behaviour 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 characteristic developed for the reduced model can thus be applied to the original model.
This paper is concerned with the study of insurance related derivatives on financial markets that are based on non-tradable underlyings, but are correlated with tradable assets. We calculate exponential utility-based indifference prices, and corresponding derivative hedges. We use the fact that they can be represented in terms of solutions of forward-backward stochastic differential equations (FBSDE) with quadratic growth generators. We derive the Markov property of such FBSDE and generalize results on the differentiability relative to the initial value of their forward components. In this case the optimal hedge can be represented by the price gradient multiplied with the correlation coefficient. This way we obtain a generalization of the classical ‘delta hedge’ in complete markets.
We consider Backward Stochastic Differential Equations (BSDEs) with generators that grow quadratically in the control variable. In a more abstract setting, we first allow both the terminal condition and the generator to depend on a vector parameter x. We give sufficient conditions for the solution pair of the BSDE to be differentiable in x. These results can be applied to systems of forward-backward SDE. If the terminal condition of the BSDE is given by a sufficiently smooth function of the terminal value of a forward SDE, then its solution pair is differentiable with respect tot the initial vector of the forward equation. Finally we prove sufficient conditions for solutions of quadratic BSDEs to be differentiable in the variational sense (Malliavin differentiable).
We consider simple models of financial markets with less and better
informed investors described by a smaller and a larger filtration on a
general stochastic basis that describes the market dynamics, including
continuous and jump components. We study the relation between different forms of non existance of arbitrage and the characteristics of the stochastic basis under the different filtrations. This is achieved through the analysis of the properties of the numéraire portfolio. Furthermore, we focus on the problem of calculating the additional logarithmic utility of the better informed investor in terms of the Shannon antropy of is additional information. The information drift, i.e. the drift to eliminate in order to preserved the martingale property in the larger filtration terms out to be the crucial quantity needed to tackle these problems. We show that the expected
ed logarithmic utility increment due to better information equals its Shannon
entropy also in case of a pure jump basis with jumps that are quadratically
hedgeable, and so extend a similar result known for bases consisting of
continuous semimartingales. An example illustrates that the equality may
not persist if both continuous and jump components are present in the
underlying.
We consider insurance derivatives depending on an external physical risk process, for example a temperature in a low dimensional climate model. We assume that this process is correlated with a tradable financial asset. We derive optimal strategies for exponential utility from terminal wealth, determine the indifference prices of the derivatives, and interpret them in terms of diversification pressure. Moreover we check the optimal investment strategies for standard admissibility criteria. Finally we compare the static risk connected with an insurance derivative to the reduced risk due to a dynamic investment into the correlated asset. We show that dynamic hedging reduces the risk aversion in terms of entropic risk measures by a factor related to the correlation.
In this paper we study BSDEs arising from a special class of backward stochastic partial differential equations (BSPDEs) that is intimately related to utility maximization problems with respect to arbitrary utility functions. After providing existence and uniqueness we discuss the numerical realizability. Then we study utility maximization problems on incomplete financial markets whose dynamics are governed by continuous semimartingales. Adapting standard methods that solve the utility maximization problem using BSDEs, we give solutions for the portfolio optimization problem which involve the delivery of a liability at maturity. We illustrate our study by numerical simulations for selected examples. As a byproduct we prove existence of a solution to a very particular quadratic growth BSDE with unbounded terminal condition. This complements results on this topic obtained in [6,7,8].
In this paper we consider the first exit problem of an overdamped
Lévy driven particle in a confining potential. We survey results
obtained in recent years from our work on the Kramers' times for
dynamical systems of this type with Lévy perturbations containing
heavy, and exponentially light jumps, and compare them to the well
known case of dynamical systems with Gaussian perturbations. It
turns out that exits induced by Lévy processes with jumps are
always essentially faster than Gaussian exits.
With an emphasis on generators with quadratic growth in the control variable we consider
measure solutions of BSDE, a solution concept corresponding to the notion of risk neutral
measure in mathematical finance. In terms of measure solutions, solving a BSDE reduces
to martingale representation with respect to an underlying filtration. Measure solutions
related to measures equivalent to the historical one provide classical solutions. We derive
the existence of measure solutions in scenarios in which the generating functions are just
continuous, of at most linear growth in the control variable (corresponding to generators of
at most quadratic growth in the usual sense), and with a random bound in the time parameter
whose stochastic integral is a BMO martingale. Our main tools include a stability property
of sequences of measure solutions, for which a limiting solution is obtained by means of the
weak convergence of measures.
We consider the problem of utility maximization for small traders on incomplete
financial markets. As opposed to most of the papers dealing with this
subject, the investors’ trading strategies we allow underly constraints described
by closed, but not necessarily convex, sets. The final wealths obtained by trading
under these constraints are identified as stochastic processes which usually are
supermartingales, and even martingales for particular strategies. These strategies
are seen to be optimal, and the corresponding value functions determined
simply by the initial values of the supermartingales. We separately treat the
cases of exponential, power and logarithmic utility.
We consider financial markets with agents exposed to an external source of
risk which cannot be hedged through investments on the capital market alone.
The sources of risk we think of may be weather and climate. Therefore we face
a typical example of an incomplete financial market. We design a model of a
market on which the external risk becomes tradable. In a first step we complete
the market by introducing an extra security which valuates the external risk
through a process parameter describing its market price. If this parameter is
fixed, risk has a price and every agent can maximize the expected exponential
utility with individual risk aversion obtained from his risk exposure on the one
hand and his investment into the financial market consisting of an exogenous set
of stocks and the insurance asset on the other hand. In the second step, the
market price of risk parameter has to be determined by a partial equilibrium
condition which just expresses the fact that in equilibrium the market is cleared
of the second security. This choice of market price of risk is performed in the
framework of nonlinear backwards stochastic differential equations.
Equilibrium trading of climate and weather risk and numerical simulation in a Markovian framework
(2004)
We consider financial markets with agents exposed to external sources of risk
caused for example by short term climate events such as the South Pacific sea
surface temperature anomalies widely known under the name El Nino. Since
such risks cannot be hedged through investments on the capital market alone,
we face a typical example of an incomplete financial market. In order to make
this risk tradable, we use a financial market model in which an additional insurance
asset provides another possibility of investment besides the usual capital
market. Given one of many possible market prices of risk each agent can maximize
his individual exponential utility from his income obtained from trading in
the capital market, the additional security, and his risk exposure function. Under
the equilibrium market clearing condition for the insurance security the market
price of risk is uniquely determined by a backward stochastic differential equation.
We translate these stochastic equations via the Feynman-Kac formalism
into semi-linear parabolic partial differential equations. Numerical schemes are
available by which these semilinear pde can be simulated. We choose two simple
qualitatively interesting models to describe sea surface temperature, and with
an ENSO risk exposed fisher and farmer and a climate risk neutral bank three
model agents with simple risk exposure functions. By simulating the expected
appreciation price of risk trading, the optimal utility of the agents as a function
of temperature, and their optimal investment into the risk trading security we
obtain first insight into the dynamics of such a market in simple situations.
We consider financial markets with two kinds of small traders: regular traders
who perceive the asset price process S through its natural filtration, and insid-
ers who possess some information advantage which makes the filtrations through
which they perceive the evolution of the market richer. The basic question we dis-
cuss is the link between (NFLVR), the semimartingale property of S viewed from
the agent’s perspective, and bounded expected utility. We show that whenever
an agent’s expected utility is finite, S is a semimartingale with a Doob-Meyer
decomposition featuring a martingale part and an information drift. The ex-
pected utility gain of an insider with respect to a regular trader is calculated in
a completely general setting. In particular, for the logarithmic utility function,
utility gain is a function of the relative information drift alone, regardless of the
completeness of the market.
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.