In this paper we consider a general optimal consumption-portfolio selection problem of an infinitely-lived agent whose consumption rate process is subject to subsistence constraints before retirement. That is, her consumption rate should be greater than or equal to some positive constant before retirement. We integrate three optimal decisions which are the optimal consumption, the optimal investment choice and the optimal stopping problem in which the agent chooses her retirement time in one model. We obtain the explicit forms of optimal policies using a martingale method and a variational inequality arising from the dual function of the optimal stopping problem. We treat the optimal retirement time as the first hitting time when her wealth exceeds a certain wealth level which will be determined by a free boundary value problem and duality approaches. We also derive closed forms of the optimal wealth processes before and after retirement. Some numerical examples are presented for the case of constant relative risk aversion (CRRA) utility class.

We study an optimal portfolio and consumption choice problem of family that combines life insurance of parents who receive deterministic labor income until fixed time horizon T . We consider utility functions of parents and children separately and assumed that parents have uncertain lifetime. If parents die before T , children have no income and they choose the optimal consumption and portfolio with remaining wealth combining the insurance benefit. Before the death time of parents, the object of family is to maximize weighted average of utility of parents and children. It is assumed that the utility function of both parents and children belong to HARA utility class. Using HARA utility we impose the condition that instantaneous consumption rate should be above a given lower bound.We analyze how the change of weight and other parameters such as lower bound of consumption, income process, hazard rate affect the optimal policies using various numerical examples. * We are grateful to Hyeng Keun Koo and Jaeyoung Sung for useful comments and advice.

Algal bloom in rivers is a major environmental concern which threatens the stable water supply and river ecosystem. Due to its complexity and nonlinearity, previous studies have tried various machine learning techniques to predict algal bloom. However, conventional approaches have limitations on predicting unobserved near future, and thus it is hard to apply to actual preparation policy. In this study, long short-term memory (LSTM), as a deep learning approach, is applied to predict the concentration of chlorophyll-a. Daily measured water quality information is used as input data and chlorophyll-a is used to output value for representing algal bloom. In addition to 1-day prediction, 4days prediction task is attempted as sequence data prediction. As a result, LSTM network shows better performance, compared to the previous approaches, in predicting chlorophyll-a in 4-days prediction as well as 1-day prediction. In addition, the regularization methods are applied to model and batch normalization is proved to be a suitable way to improve accuracy. This result can lead to improvement in preventing algal bloom and also suggest various applications of deep learning methods in chlorophyll-a prediction task.

SUMMARYIn the recent works (Commun. Numer. Meth. Engng 2001; 17:881; to appear), the superiority of the non-linear transformations containing a real parameter b = 0 has been demonstrated in numerical evaluation of weakly singular integrals. Based on these transformations, we define a so-called parametric sigmoidal transformation and employ it to evaluate the Cauchy principal value and Hadamard finitepart integrals by using the Euler-Maclaurin formula. Better approximation is expected due to the prominent properties of the parametric sigmoidal transformation of whose local behaviour near x = 0 is governed by parameter b.Through the asymptotic error analysis of the Euler-Maclaurin formula using the parametric sigmoidal transformation, we can observe that it provides a distinct improvement on its predecessors using traditional sigmoidal transformations. Numerical results of some examples show the availability of the present method.

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A method for stable numerical differentiation of noisy data is proposed. The method requires solving a Volterra integral equation of the second kind. This equation is solved analytically. In the examples considered its solution is computed analytically. Some numerical results of its application are presented. These examples show that the proposed method for stable numerical differentiation is numerically more efficient than some other methods, in particular, than variational regularization.

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We propose and analyze the spectral collocation approximation for the partial integrodifferential equations with a weakly singular kernel. The space discretization is based on the pseudo-spectral method, which is a collocation method at the Gauss-Lobatto quadrature points. We prove unconditional stability and obtain the optimal error bounds which depend on the time step, the degree of polynomial and the Sobolev regularity of the solution.