3,999 research outputs found
Branching processes, the max-plus algebra and network calculus
Branching processes can describe the dynamics of various queueing systems, peer-to-peer systems, delay tolerant networks, etc. In this paper we study the basic stochastic recursion of multitype branching processes, but in two non-standard contexts. First, we consider this recursion in the max-plus algebra where branching corresponds to finding the maximal offspring of the current generation. Secondly, we consider network-calculus-type deterministic bounds as introduced by Cruz, which we extend to handle branching-type processes. The paper provides both qualitative and quantitative results and introduces various applications of (max-plus) branching processes in queueing theory
Cross sectional efficient estimation of stochastic volatility short rate models
We consider the problem of estimation of term structure of interest rates. Filtering theory approach is very natural here with the underlying setup being non-linear and non-Gaussian. Earlier works make use of Extended Kalman Filter (EKF). However, the EKF in this situation leads to inconsistent estimation of parameters, though without high bias. One way to avoid this is to use methods like Efficient Method of Moments or Indirect Inference Method. These methods, however, are numerically very demanding. We use Kitagawa type scheme for nonlinear filtering problem, which solves the inconsistency problem without being numerically so demanding. \u
Can local particle filters beat the curse of dimensionality?
The discovery of particle filtering methods has enabled the use of nonlinear
filtering in a wide array of applications. Unfortunately, the approximation
error of particle filters typically grows exponentially in the dimension of the
underlying model. This phenomenon has rendered particle filters of limited use
in complex data assimilation problems. In this paper, we argue that it is often
possible, at least in principle, to develop local particle filtering algorithms
whose approximation error is dimension-free. The key to such developments is
the decay of correlations property, which is a spatial counterpart of the much
better understood stability property of nonlinear filters. For the simplest
possible algorithm of this type, our results provide under suitable assumptions
an approximation error bound that is uniform both in time and in the model
dimension. More broadly, our results provide a framework for the investigation
of filtering problems and algorithms in high dimension.Comment: Published at http://dx.doi.org/10.1214/14-AAP1061 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Concentration Bounds for Stochastic Approximations
We obtain non asymptotic concentration bounds for two kinds of stochastic
approximations. We first consider the deviations between the expectation of a
given function of the Euler scheme of some diffusion process at a fixed
deterministic time and its empirical mean obtained by the Monte-Carlo
procedure. We then give some estimates concerning the deviation between the
value at a given time-step of a stochastic approximation algorithm and its
target. Under suitable assumptions both concentration bounds turn out to be
Gaussian. The key tool consists in exploiting accurately the concentration
properties of the increments of the schemes. For the first case, as opposed to
the previous work of Lemaire and Menozzi (EJP, 2010), we do not have any
systematic bias in our estimates. Also, no specific non-degeneracy conditions
are assumed.Comment: 14 page
Ruin probabilities in a finite-horizon risk model with investment and reinsurance
A finite horizon insurance model is studied where the risk/reserve process can be controlled by reinsurance and investment in the financial market. Obtaining explicit optimal solutions for the minimizing ruin probability problem is a difficult task. Therefore, we consider an alternative method commonly used in ruin theory, which consists in deriving inequalities that can be used to obtain upper bounds for the ruin probabilities and then choose the control to minimize the bound. We finally specialize our results to the particular, but relevant, case of exponentially distributed claims and compare for this case our bounds with the classical Lundberg bound.Risk process, Reinsurance and investment, Lundberg’s inequality, 91B30, 93E20, 60J28
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