In this paper we consider the optimal dividend problem for an insurance company whose risk process evolves as a spectrally negative L\'{e}vy process in the absence of dividend payments. The classical dividend problem for an insurance company consists in finding a dividend payment policy that maximizes the total expected discounted dividends. Related is the problem where we impose the restriction that ruin be prevented: the beneficiaries of the dividends must then keep the insurance company solvent by bail-out loans. Drawing on the fluctuation theory of spectrally negative L\'{e}vy processes we give an explicit analytical description of the optimal strategy in the set of barrier strategies and the corresponding value function, for either of the problems. Subsequently we investigate when the dividend policy that is optimal among all admissible ones takes the form of a barrier strategy.

On the optimal dividend problem for a spectrally negative L\'{e}vy process

Lévy Processes” (eight papers), “III. Empirical Processes” (four papers), and “IV. Stochastic Differential Equations” (four papers). Here are some comments about the individual papers: In I.2 (paper 2 of Part I) the covariance representation method, which relies on Clark’s formula on path spaces, is used to obtain concentration inequalities for functionals of Brownian motion on a manifold, allowing one to obtain tail estimates for this Brownian motion. In I.4 a transportation inequality for the canonical Gaussian measure in R is obtained and applied to Khintchine–Kahane inequalities for norms of random series with nonsymmetric Bernoulli coefficients. In II.1 exponential inequalities for U -statistics of order two are presented; these rely upon the Talagrand inequality for empirical processes but also use martingale type inequalities. In II.2 the unconditional convergence of a Gaussian [and, more generally, independent, identically distributed (iid)] series in a Banach space is studied. Applications to Karhunen–Love representations of Gaussian processes are given. In II.3 estimates of tail properties and moments of multidimensional chaos generated by positive random variables with log concave tails are given. In II.4 a quantitative technique for studying the asymptotic distribution of sequences of Markov processes in infinite dimensions is proposed. The proof relies on the properties of an associated sequence of exponential martingales. In II.5 it is shown that a moving average process driven by a symmetric Lévy process and with a kernel with finite total 2-variation admits an almost surely bounded version. In II.6 a Markovian approach to the entropic convergence in the central limit theorem is presented. The emphasis is on the speed of convergence, as well as relaxing a spectral gap assumption. In II.7 a new version of the Khintchine–Kahane inequality for general Bernoulli random variables is presented with the help of hypercontractive methods. In III.1 necessary and sufficient conditions for the moderate deviations of empirical processes and sums of iid random vectors on a separable Banach space are given. In III.2 exponential concentration inequalities for subadditive functions of independent random variables are obtained. As a consequence, Talagrand’s inequality for empirical processes is refined thanks to further developments of the entropy method introduced by M. Ledoux. In III.3 ratio limit theorems for empirical processes are obtained with the help of concentration inequalities. In III.4 asymptotic distributions of trimmed Wasserstein distances between the true and the empirical distribution function are obtained via weighted approximation results for uniform empirical processes. In IV.1 sharp rates of convergence for splitting-up approximations of stochastic partial differential equations are obtained. The error is estimated in terms of Sobolev’s norm. In IV.4 the existence and uniqueness of a strong solution for a stochastic differential equation driven by a fractional Brownian motion with Hurst index H < 1/2 and with a possibly time-dependent drift which satisfies a suitable integrability condition is obtained. In short, the book presents a wide variety of inequalities which are established for either stochastic processes or sequences of random variables. In 2006 this is still an active domain of research in transportation problems.

Statistical Inference for Ergodic Diffusion Processes

In this paper, an energy management issue is considered for energy Internet where microgrids (MGs) are interconnected via energy routers (ERs). Focusing on an individual MG, we propose controllers in microturbines (MTs) and the ER, such that the following three criteria are hold simultaneously. First, a bottom-up energy management approach is realized. Second, the operation cost of utilizing battery energy storage devices is minimized. Third, the situation of overcontrol with respect to MTs is considered to be avoided. Besides, we develop a novel hybrid modeling method combining both recurrent neural networks and Ornstein–Uhlenbeck process to obtain accurate power models for both photovoltaic panels and loads. Next, we formulate our energy management issue into a stochastic optimal control problem and solve it via dynamic programming approach. Finally, examples illustrating the feasibility of the proposed methods are provided.

Stochastic Optimal Control for Energy Internet: A Bottom-Up Energy Management Approach

The development of energy Internet (EI) makes it possible to achieve better utilization of distributed renewable energy sources with the power sharing functionality introduced by energy routers (ERs). In this paper, a bottom-up EI architecture is designed, and a novel data-driven dynamical control strategy is proposed. Intelligent controllers augmented by deep reinforcement learning (DRL) techniques are adopted for the operation of each microgrid independently in the bottom layer. Moreover, the concept of curriculum learning (CL) is integrated into DRL to improve the sample efficiency and accelerate the training process. Based on the power exchange plan determined in the bottom layer, considering the stochastic nature of electricity price in the future power market, the optimal power dispatching scheme in the upper layer is decided via model predictive control. The simulation has shown that, under the bottom-up architecture, compared with the conventional methods such as proportional integral and optimal power flow, the proposed method reduces overall generation cost by more than 7.1% and 37%. Meanwhile, the introduced CL-based training strategy can significantly speed up the convergence during the training of DRL. Last but not least, our method increases the profit of energy trading between ERs and the main grid.

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Data-driven Dynamical Control for Bottom-up Energy Internet System

The Ornstein–Uhlenbeck process with reflection, which has been the subject of an enormous body of literature, both theoretical and applied, is a process that returns continuously and immediately to the interior of the state space when it attains a certain boundary. In this work, we are mainly concerned with the study of the asymptotic behavior of the trajectory fitting estimator for nonergodic reflected Ornstein–Uhlenbeck processes, including strong consistency and asymptotic distribution. Moreover, we also prove that this kind of estimator for ergodic reflected Ornstein–Uhlenbeck processes does not possess the property of strong consistency.

Asymptotic Behaviour of the Trajectory Fitting Estimator for Reflected Ornstein–Uhlenbeck Processes

A reflected Ornstein–Uhlenbeck process is a process that returns continuously and immediately to the interior of the state space when it attains a certain boundary. It is an extended model of the traditional Ornstein–Uhlenbeck process being extensively used in finance as a one-factor short-term interest rate model. Under some mild conditions, this paper is devoted to the study of the analogue of the Cramer–Rao lower bound of a general class of parameter estimation of the unknown parameter in reflected Ornstein–Uhlenbeck processes.

A general lower bound of parameter estimation for reflected Ornstein-Uhlenbeck processes

Reflected Ornstein-Uhlenbeck process is a process that returns continuously and immediately to the interior of the state space when it attains a certain boundary. It is an extended model of the traditional Ornstein-Uhlenbeck process being extensively used in finance as a one-factor short-term interest rate model. In this paper, under certain constraints, we are concerned with the problem of estimating the unknown parameter in the reflected Ornstein-Uhlenbeck processes with the general drift coefficient. The methodology of estimation is built upon the maximum likelihood approach and the method of stochastic integration. The strong consistency and asymptotic normality of estimator are derived. As a by-product of the use, we also establish Girsanovs theorem of our model in this paper.

Qing-Pei Zang

Papers

Asymptotic Behaviour of the Trajectory Fitting Estimator for Reflected Ornstein–Uhlenbeck Processes

A general lower bound of parameter estimation for reflected Ornstein-Uhlenbeck processes

Parameter estimation for generalized diffusion processes with reflected boundary