We consider the problem of optimal singular control of a stochastic partial differential equation (SPDE) with space-mean dependence. Such systems are proposed as models for population growth in a random environment. We obtain sufficient and necessary maximum principles for such control problems. The corresponding adjoint equation is a reflected backward stochastic partial differential equation (BSPDE) with space-mean dependence. We prove existence and uniqueness results for such equations. As an application we study optimal harvesting from a population modelled as an SPDE with space-mean dependence.
An estimate uniform in time for the transition probability of di®usion processes with small drift is given. This also covers the case of a degenerate di®usion describing a stochastic perturbation of a particle moving according to the Newton's law. Moreover the random wave operator for such a particle is described and the analogue of asymptotic completeness is proven, the latter in the case of a su±ciently small drift.
We analyze the asymptotic behaviour of the heat kernel dened by a stochastically perturbedgeodesic ow on the cotangent bundle of a Riemannian manifold for small time and smalldiusion parameter. This extends WKB-type methods to a particular case of a degenerateHamiltonian. We give uniform bounds for the solution of the degenerate Hamiltonian boundaryvalue problem for small time. The results are exploited to derive two sided estimates andmultiplicative asymptotics for the heat kernel semigroup and its trace.
We derive a Smoluchowski-Kramers type scaling limit for second order stochastic differential equations driven by Fractional Brownian motion.We show a Girsanov theorem for the solution processes with respect to corresponding Fractional Ornstein-Uhlenbeck processes which are Gaussian. This reveals existence of weak solutions as well as a weak scaling limit. Subsequently the results are strengthened.
Brownian motion has been constructed in different ways. Einstein was the most outstanding physicists involved in its construction. From a physical point of view a dynamical theory of Brownian motion was favorable. The Ornstein-Uhlenbeck process models such a dynamical theory and E. Nelson amongst others derived Brownian motion from Ornstein-Uhlenbeck theory via a scaling limit. In this paper we extend the scaling result to α-stable Lévy processes.
We establish a scaling limit for autonomous stochastic Newton equations, the solutions are often called nonlinear stochastic oscillators,where the nonlinear drift includes a mean field term of Mckean type and the driving noise is Gaussian. Uniform convergence in sense is achieved by applying -type estimates and the Gronwall Theorem.The approximation is also called Smoluchowski-Kramers limit and is a particular averaging technique studied by Papanicolaou. It reveals an approximation of diffusions with a mean-field contribution in the drift by diffusions with differentiable trajectories.
We study a second order stochastic differential equation which was derived from a non-linear Schrödinger equation with non-linear damping and additive noise to describe the width of related wave solutions. We prove existence of solutions and discuss their long-time behaviour.
We show that a physically motivated trial solution of a dampeddriven non-linear Schr ̈odinger equation does neither encounter collapse norso-called pseudocollapse although the exponent of the non-linearity is crit-ical. This result sheds new light on the accuracy of numerical solutions tothis problem obtained in an earlier paper where the authors claim pseudo-collapse of the trial solution when the variance of the driving noise is below a certain level.
We consider the focusing 2D nonlinear Schrodinger equation, perturbed by a damping term, and driven by multiplicative noise. We show that a physically motivated trial solution does not collapse for any admissible initial condition although the exponent of the nonlinearity is critical. Our method is based on the construction of a global solution to a singular stochastic Hamiltonian system used to connect trial solution and Schrodinger equation.
Weshow that paradoxical consequences of violations of Bell's inequality areinduced by the use of an unsuitable probabilistic description forthe EPR-Bohm-Bell experiment. The conventional description (due to Bell) isbased on a combination of statistical data collected for differentsettings of polarization beam splitters (PBSs). In fact, such dataconsists of some conditional probabilities which only partially define aprobability space. Ignoring this conditioning leads to apparent contradictions inthe classical probabilistic model (due to Kolmogorov). We show howto make a completely consistent probabilistic model by taking intoaccount the probabilities of selecting the settings of the PBSs.Our model matches both the experimental data and is consistentwith classical probability theory.
We investigate mean-field games from the point of view of a large number of indistinguishable players, which eventually converges to infinity. The players are weakly coupled via their empirical measure. The dynamics of the states of the individual players is governed by a non-autonomous pure jump type semi group in a Euclidean space, which is not necessarily smoothing. Investigations are conducted in the framework of non-linear Markovian semi groups. We show that the individual optimal strategy results from a consistent coupling of an optimal control problem with a forward non-autonomous dynamics. In the limit as the number N of players goes to infinity this leads to a jump-type analog of the well-known non-linear McKean–Vlasov dynamics. The case where one player has an individual preference different from the ones of the remaining players is also covered. The two results combined reveal an epsilon-NashEquilibrium for the N-player games.
We investigate mean field games from the point of view of a large number of indistinguishable players which eventually converges to in- finity. The players are weakly coupled via their empirical measure. The dynamics of the individual players is governed by pure jump type propagators over a finite space. Investigations are conducted in the framework of non-linear Markov processes. We show that the individual optimal strategy results from a consistent coupling of an optimal control problem with a forward non-autonomous dynamics. In the limit as the number N of players goes to infinity this leads to a jump-type analog of the well-known non-linear McKean-Vlasov dynamics. The case where one player has an individual preference different from the ones of the remaining players is also covered. The two results combined reveal a 1 N -Nash Equilibrium for the approximating system of N players.
In this paper, we investigate reflected backward doubly stochastic differential equations (RBDSDEs) with a lower not necessarily right-continuous obstacle. First, we establish the existence and uniqueness of a solution to RBDSDEs with Lipschitz drivers. In the second part, we present a comparison theorem and we prove the existence of a minimal solution to the RBDSDE with the continuous driver.
We study a version of the functional Hodrick-Prescott filter where the associated operator is not necessarily compact, but merely closed and densely defined with closed range. We show that the associate doptimal smoothing operator preserves the structure obtained in the compact case, when the underlying distribution of the data is Gaussian.
Detecting structural breaks is an essential task for the statistical analysis of time series, for example, for fitting parametric models to it. In short, structural breaks are points in time at which the behaviour of the time series substantially changes. Typically, no solid background knowledge of the time series under consideration is available. Therefore, a black-box optimization approach is our method of choice for detecting structural breaks. We describe a genetic algorithm framework which easily adapts to a large number of statistical settings. To evaluate the usefulness of different crossover and mutation operations for this problem, we conduct extensive experiments to determine good choices for the parameters and operators of the genetic algorithm. One surprising observation is that use of uniform and one-point crossover together gave significantly better results than using either crossover operator alone. Moreover, we present a specific fitness function which exploits the sparse structure of the break points and which can be evaluated particularly efficiently. The experiments on artificial and real-world time series show that the resulting algorithm detects break points with high precision and is computationally very efficient. A reference implementation with the data used in this paper is available as an applet at the following address: http://www.imm.dtu.dk/~pafi/TSX/. It has also been implemented as package SBRect for the statistics language R.
Detecting structural breaks is an essential task for the statistical analysis of time series, for example, for fitting parametric models to it. In short, structural breaks are points in time at which the behavior of the time series changes. Typically, no solid background knowledge of the time series under consideration is available. Therefore, a black-box optimization approach is our method of choice for detecting structural breaks. We describe a \ea framework which easily adapts to a large number of statistical settings. The experiments on artificial and real-world time series show that the algorithm detects break points with high precision and is computationally very efficient.
A reference implementation is availble at the following address:
http://www2.imm.dtu.dk/\~\/pafi/SBX/launch.html
In this paper, we generalize the concept of gamma bridge in the sense that the length will be random, that is, the time to reach the given level is random. The main objective of this paper is to show that certain basic properties of gamma bridges with deterministic length stay true also for gamma bridges with random length. We show that the gamma bridge with random length is a pure jump process and that its jumping times are countable and dense in the random interval bounded by 0 and the random length. Moreover, we prove that this process is a Markov process with respect to its completed natural filtration as well as with respect to the usual augmentation of this filtration, which leads us to conclude that its completed natural filtration is right continuous. Finally, we give its canonical decomposition with respect to the usual augmentation of its natural filtration.
A fundamental task in the analysis of time series is to detect structural breaks. A break indicates a significant change in the behaviour of the series. One method to formalise the notion of a break point, is to fit statistical models piecewise to the series. To find break points, the endpoints of the pieces are varied as is their number. A structural break is indicated by a significant change of the model parameters in adjacent pieces. Both, varying the pieces and repeatedly fitting models to them, are usually computationally very expensive. By combining genetic algorithms with a preprocessing of the time series we design a very fast algorithm for structural break detection. It reduces the time for model-fitting from linear to logarithmic in the length of the series. We show how this method can be used to find structural breaks for time series which are piecewise generated by AR(p)-models. Moreover, we introduce a nonparametric model for which the speed-up can also be achieved. Additionally we briefly present simulation results which demonstrate the manifold applications of these methods. A reference implementation is available at http://www2.imm.dtu.dk/~pafi/StructBreak/index.html
We introduce a platform which supplies an easy-to-handle, interactive, extendable,and fast analysis tool for time series analysis. In contrast to other software suits like Maple,Matlab, or R, which use a command-line-like interface and where the user has to memorize/look-up the appropriate commands, our application is select-and-click-driven. It allows to derive manydierent sequences of deviations for a given time series and to visualize them in dierent waysin order to judge their expressive power and to reuse the procedure found.For many transformations or model-ts, the user may choose between manual and automatedparameter selection. The user can dene new transformations and add them to the system. Theapplication contains ecient implementations of advanced and recent techniques for time seriesanalysis including techniques related to extreme value analysis and ltering theory. It has beensuccessfully applied to time series in economics, e.g. reinsurance, and to vibrational stressdata for machinery. The software is web-deployed, but runs on the user's machine, allowingto process sensitive data locally without having to send it away. The software can be accessedunder http://www.imm.dtu.dk/~paf/TSA/launch.html.
Nagel, Stein, and Wainger have given a uniform estimate for thevolume of sub-Riemannianballs associated with an operator given as sum of squares of vector fields satisfying a uniform strongH{\"o}rmander hypothesis. L\'eandre has given nonuniform estimates forthe volume of ballsassociated with an operator L= 1/2 sum X_i^2 +X_0, and has shown that theBismut conditionis important to establish the relation between hypoelliptic kernels andvolumes of balls. In this paper we settle the uniform case for ballsassociated with distances where the Bismut condition is involved.