Efficient pointwise estimation based on discrete data in ergodic nonparametric diffusions

A truncated sequential procedure is constructed for estimating the drift coefficient at a given state point based on discrete data of ergodic diffusion process. A nonasymptotic upper bound is obtained for a pointwise absolute error risk. The optimal convergence rate and a sharp constant in the bounds are found for the asymptotic pointwise minimax risk. As a consequence, the efficiency is obtained of the proposed sequential procedure.Comment: Published at http://dx.doi.org/10.3150/14-BEJ655 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Oracle inequalities for the stochastic differential equations

International audienceThis paper is a survey of recent results on the adaptive robust non parametric methods for the continuous time regression model with the semi-martingale noises with jumps. The noises are modeled by the Lévy processes, the Ornstein-Uhlenbeck processes and semi-Markov processes. We represent the general model selection method and the sharp oracle inequalities methods which provide the robust efficient estimation in the adaptive setting. Moreover, we present the recent results on the improved model selection methods for the nonparametric estimation problems

Задача оптимального инвестирования и потребления для спредовых рынков состохастической волатильностью

We consider a spread financial market defined by the Ornstein–Uhlenbeck (OU) process with a diffusion coefficient driven by a stochastic differential equation.For this market we study the optimal consumption/investment problem under logarithmic utilities. This problem is studied on the base of the stochastic dynamical programming approach.To this end we show aspecial verification theorem for this case.Then,we study the corresponding Hamilton–Jacobi–Bellman (HJB) equation and find its solution in explicit form.Finally,through this solution we construct the optimal financial strategies

Optimal investment and consumption for Ornstein-Uhlenbeck spread financial markets with power utility

We consider a spread financial market defined by the Ornstein-Uhlenbeck (OU) process. We construct the optimal consumption/investment strategy for the power utility function. We study the Hamilton-Jacobi-Bellman (HJB) equation by the Feynman-Kac (FK) representation. We show the existence and uniqueness theorem for the classical solution. We study the numeric approximation and we establish the convergence rate. It turns out that in this case the convergence rate for the numerical scheme is super geometrical, i.e., more rapid than any geometrical on

Задачи оптимизации портфеля при логарифмических функциях полезности

Рассматриваются задачи оптимального потребления иинвестирования для финансовых рынков,описываемых семимартингалами с независимыми приращениями при логарифмических полезностях.Найдены оптимальные финансовые стратегии в явном виде.Затем с помощью подхода Леланда-Лепинетта полученные стратегии модифицируются для учета транзакционных издержекна финансовых рынках.Болеет ого,показано,что такие финансовые портфели обеспечивают асимптотически оптимальные инвестиции и потребление для рынков странзакционными издержками,когда количество ревизий портфеля стремится к бесконечности

Оптимальное инвестирование и потребление для степенных функций полезности

We consider a portfolio optimization problem for financial markets driven by Levy processes with non constant coefficients. For power utility functions we find the optimal strategy in explicit form. Moreover, using this strategy and the Leland approach we develop asymptotic optimal investment and consumption methods for the financial markets with proportional transaction costs