12,793 research outputs found
On certain integral functionals of squared Bessel processes
Let be a squared Bessel process. Following a Feynman-Kac approach, the
Laplace transforms of joint laws of are studied
where is the first hitting time of by and is a random
variable measurable with respect to the history of until . A subset of
these results are then used to solve the associated small ball problems for
and determine a Chung's law of iterated logarithm.
is also considered as a purely discontinuous
increasing Markov process and its infinitesimal generator is found. The
findings are then used to price a class of exotic derivatives on interest rates
and determine the asymptotics for the prices of some put options that are only
slightly in-the-money
Monte Carlo optimization of decentralized estimation networks over directed acyclic graphs under communication constraints
Motivated by the vision of sensor networks, we consider decentralized estimation networks over bandwidthâlimited communication links, and are particularly interested in the tradeoff between the estimation accuracy and the cost of communications due to, e.g., energy consumption. We employ a class of inânetwork processing strategies that admits directed acyclic graph representations and yields a tractable Bayesian risk that comprises the cost of communications and estimation error penalty. This perspective captures a broad range of possibilities for processing under network constraints and enables a rigorous design problem in the form of constrained optimization. A similar scheme and the structures exhibited by the solutions have been previously studied in the context of decentralized detection. Under reasonable assumptions, the optimization can be carried out in a message passing fashion. We adopt
this framework for estimation, however, the corresponding optimization scheme involves integral operators that cannot be evaluated exactly in general. We develop an approximation framework using Monte Carlo methods and obtain
particle representations and approximate computational schemes for both the inânetwork processing strategies and their optimization. The proposed Monte Carlo optimization procedure operates in a scalable and efficient fashion and,
owing to the non-parametric nature, can produce results for any distributions provided that samples can be produced from the marginals. In addition, this approach exhibits graceful degradation of the estimation accuracy asymptotically
as the communication becomes more costly, through a parameterized Bayesian risk
Markovian Nash equilibrium in financial markets with asymmetric information and related forward-backward systems
This paper develops a new methodology for studying continuous-time Nash
equilibrium in a financial market with asymmetrically informed agents. This
approach allows us to lift the restriction of risk neutrality imposed on market
makers by the current literature. It turns out that, when the market makers are
risk averse, the optimal strategies of the agents are solutions of a
forward-backward system of partial and stochastic differential equations. In
particular, the price set by the market makers solves a nonstandard "quadratic"
backward stochastic differential equation. The main result of the paper is the
existence of a Markovian solution to this forward-backward system on an
arbitrary time interval, which is obtained via a fixed-point argument on the
space of absolutely continuous distribution functions. Moreover, the
equilibrium obtained in this paper is able to explain several stylized facts
which are not captured by the current asymmetric information models.Comment: Published at http://dx.doi.org/10.1214/15-AAP1138 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Markov bridges: SDE representation
Let be a Markov process taking values in with continuous
paths and transition function . Given a measure on
, a Markov bridge starting at
and ending at for has the law of the original process
starting at at time and conditioned to have law at time . We
will consider two types of conditioning: a) {\em weak conditioning} when
is absolutely continuous with respect to and b) {\em strong
conditioning} when for some . The main
result of this paper is the representation of a Markov bridge as a solution to
a stochastic differential equation (SDE) driven by a Brownian motion in a
diffusion setting. Under mild conditions on the transition density of the
underlying diffusion process we establish the existence and uniqueness of weak
and strong solutions of this SDE.Comment: A missing reference is adde
Point process bridges and weak convergence of insider trading models
We construct explicitly a bridge process whose distribution, in its own
filtration, is the same as the difference of two independent Poisson processes
with the same intensity and its time 1 value satisfies a specific constraint.
This construction allows us to show the existence of Glosten-Milgrom
equilibrium and its associated optimal trading strategy for the insider. In the
equilibrium the insider employs a mixed strategy to randomly submit two types
of orders: one type trades in the same direction as noise trades while the
other cancels some of the noise trades by submitting opposite orders when noise
trades arrive. The construction also allows us to prove that Glosten-Milgrom
equilibria converge weakly to Kyle-Back equilibrium, without the additional
assumptions imposed in \textit{K. Back and S. Baruch, Econometrica, 72 (2004),
pp. 433-465}, when the common intensity of the Poisson processes tends to
infinity
Monte Carlo optimization approach for decentralized estimation networks under communication constraints
We consider designing decentralized estimation schemes over bandwidth limited communication links with a particular interest in the tradeoff between the estimation accuracy and the cost of communications due to, e.g., energy
consumption. We take two classes of inânetwork processing strategies into account which yield graph representations through modeling the sensor platforms as the vertices and the communication links by edges as well as a tractable
Bayesian risk that comprises the cost of transmissions and penalty for the estimation errors. This approach captures a broad range of possibilities for âonlineâ processing of observations as well as the constraints imposed and enables a rigorous design setting in the form of a constrained optimization problem. Similar schemes as well as the structures exhibited by the solutions to the design problem has been studied previously in the context of decentralized detection. Under reasonable assumptions, the optimization can be carried out in a message passing fashion. We adopt this framework for estimation, however, the corresponding optimization schemes involve integral operators that cannot
be evaluated exactly in general. We develop an approximation framework using Monte Carlo methods and obtain particle representations and approximate computational schemes for both classes of inânetwork processing strategies
and their optimization. The proposed Monte Carlo optimization procedures operate in a scalable and efficient fashion and, owing to the non-parametric nature, can produce results for any distributions provided that samples can be
produced from the marginals. In addition, this approach exhibits graceful degradation of the estimation accuracy asymptotically as the communication becomes more costly, through a parameterized Bayesian risk
- âŠ