865 research outputs found
Optimal life insurance purchase, consumption and investment on a financial market with multi-dimensional diffusive terms
We introduce an extension to Merton’s famous continuous time model of optimal
consumption and investment, in the spirit of previous works by Pliska and Ye, to allow for
a wage earner to have a random lifetime and to use a portion of the income to purchase life
insurance in order to provide for his estate, while investing his savings in a financial market
comprised of one risk-free security and an arbitrary number of risky securities driven by multidimensional
Brownian motion. We then provide a detailed analysis of the optimal consumption,
investment, and insurance purchase strategies for the wage earner whose goal is to maximize
the expected utility obtained from his family consumption, from the size of the estate in the
event of premature death, and from the size of the estate at the time of retirement. We use
dynamic programming methods to obtain explicit solutions for the case of discounted constant
relative risk aversion utility functions and describe new analytical results which are presented
together with the corresponding economic interpretations.We thank the Calouste Gulbenkian Foundation, PRODYN-ESF, POCTI and POSI by FCT and Ministerio da Ciencia, Tecnologia e Ensino Superior, CEMAPRE, LIAAD-INESC Porto LA, Centro de Matematica da Universidade do Minho and Centro de Matematica da Universidade do Porto for their financial support. D. Pinheiro's research was supported by FCT - Fundacao para a Ciencia e Tecnologia program 'Ciencia 2007' and project 'Randomness in Deterministic Dynamical Systems and Applications' (PTDC/MAT/105448/2008). I. Duarte's research was supported by FCT - Fundacao para a Ciencia e Tecnologia grant with reference SFRH/BD/33502/2008
Relative Value Iteration for Stochastic Differential Games
We study zero-sum stochastic differential games with player dynamics governed
by a nondegenerate controlled diffusion process. Under the assumption of
uniform stability, we establish the existence of a solution to the Isaac's
equation for the ergodic game and characterize the optimal stationary
strategies. The data is not assumed to be bounded, nor do we assume geometric
ergodicity. Thus our results extend previous work in the literature. We also
study a relative value iteration scheme that takes the form of a parabolic
Isaac's equation. Under the hypothesis of geometric ergodicity we show that the
relative value iteration converges to the elliptic Isaac's equation as time
goes to infinity. We use these results to establish convergence of the relative
value iteration for risk-sensitive control problems under an asymptotic
flatness assumption
Use of approximations of Hamilton-Jacobi-Bellman inequality for solving periodic optimization problems
We show that necessary and sufficient conditions of optimality in periodic
optimization problems can be stated in terms of a solution of the corresponding
HJB inequality, the latter being equivalent to a max-min type variational
problem considered on the space of continuously differentiable functions. We
approximate the latter with a maximin problem on a finite dimensional subspace
of the space of continuously differentiable functions and show that a solution
of this problem (existing under natural controllability conditions) can be used
for construction of near optimal controls. We illustrate the construction with
a numerical example.Comment: 29 pages, 2 figure
Time scales and exponential trends to equilibrium: Gaussian model problems
We review results on the exponential convergence of multi- dimensional Ornstein-Uhlenbeck processes and discuss related notions of characteristic timescales with concrete model systems. We focus, on the one hand, on exit time distributions and provide ecplicit expressions for the exponential rate of the distribution in the small noise limit. On the other hand, we consider relaxation timescales of the process to its equi- librium measured in terms of relative entropy and discuss the connection with exit probabilities. Along these lines, we study examples which il- lustrate specific properties of the relaxation and discuss the possibility of deriving a simulation-based, empirical definition of slow and fast de- grees of freedom which builds upon a partitioning of the relative entropy functional in conjuction with the observed relaxation behaviour
The impact of current CH4 and N2O atmospheric loss process uncertainties on calculated ozone abundances and trends
The atmospheric loss processes of N2O and CH4, their estimated uncertainties, lifetimes, and impacts on ozone abundance and long-term trends are examined using atmospheric model calculations and updated kinetic and photochemical parameters and uncertainty factors from SPARC [2013]. The uncertainty ranges in calculated N2O and CH4 global lifetimes computed using the SPARC estimated uncertainties are reduced by nearly a factor of two compared with uncertainties from Sander et al. [2011]. Uncertainties in CH4 loss due to reaction with OH and O(1D) have relatively small impacts on present day global total ozone (±0.2-0.3%). Uncertainty in the Cl + CH4 reaction affects the amount of chlorine in radical vs. reservoir forms and has a modest impact on present day SH polar ozone (~±6%), and on the rate of past ozone decline and future recovery. Uncertainty in the total rate coefficient for the O(1D) + N2O reaction results in a substantial range in present day stratospheric odd nitrogen (±20-25%) and global total ozone (±1.5-2.5%). Uncertainty in the O(1D) + N2O reaction branching ratio for the O2 + N2 and 2*NO product channels results in moderate impacts on odd nitrogen (±10%) and global ozone (±1%),with uncertainty in N2O photolysis resulting in relatively small impacts (±5% in odd nitrogen, ±0.5% in global ozone). Uncertainties in the O(1D) + N2O reaction and its branching ratio also affect the rate of past global total ozone decline and future recovery, with a range in future ozone projections of ±1-1.5% by 2100, relative to present day
On Exceptional Times for generalized Fleming-Viot Processes with Mutations
If is a standard Fleming-Viot process with constant mutation rate
(in the infinitely many sites model) then it is well known that for each
the measure is purely atomic with infinitely many atoms. However,
Schmuland proved that there is a critical value for the mutation rate under
which almost surely there are exceptional times at which is a
finite sum of weighted Dirac masses. In the present work we discuss the
existence of such exceptional times for the generalized Fleming-Viot processes.
In the case of Beta-Fleming-Viot processes with index we
show that - irrespectively of the mutation rate and - the number of
atoms is almost surely always infinite. The proof combines a Pitman-Yor type
representation with a disintegration formula, Lamperti's transformation for
self-similar processes and covering results for Poisson point processes
A Semi-Lagrangian scheme for a modified version of the Hughes model for pedestrian flow
In this paper we present a Semi-Lagrangian scheme for a regularized version
of the Hughes model for pedestrian flow. Hughes originally proposed a coupled
nonlinear PDE system describing the evolution of a large pedestrian group
trying to exit a domain as fast as possible. The original model corresponds to
a system of a conservation law for the pedestrian density and an Eikonal
equation to determine the weighted distance to the exit. We consider this model
in presence of small diffusion and discuss the numerical analysis of the
proposed Semi-Lagrangian scheme. Furthermore we illustrate the effect of small
diffusion on the exit time with various numerical experiments
Systemic Risk and Default Clustering for Large Financial Systems
As it is known in the finance risk and macroeconomics literature,
risk-sharing in large portfolios may increase the probability of creation of
default clusters and of systemic risk. We review recent developments on
mathematical and computational tools for the quantification of such phenomena.
Limiting analysis such as law of large numbers and central limit theorems allow
to approximate the distribution in large systems and study quantities such as
the loss distribution in large portfolios. Large deviations analysis allow us
to study the tail of the loss distribution and to identify pathways to default
clustering. Sensitivity analysis allows to understand the most likely ways in
which different effects, such as contagion and systematic risks, combine to
lead to large default rates. Such results could give useful insights into how
to optimally safeguard against such events.Comment: in Large Deviations and Asymptotic Methods in Finance, (Editors: P.
Friz, J. Gatheral, A. Gulisashvili, A. Jacqier, J. Teichmann) , Springer
Proceedings in Mathematics and Statistics, Vol. 110 2015
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