On the rates of convergence of simulation based optimization algorithms for optimal stopping problems

In this paper we study simulation based optimization algorithms for solving discrete time optimal stopping problems. This type of algorithms became popular among practioneers working in the area of quantitative finance. Using large deviation theory for the increments of empirical processes, we derive optimal convergence rates and show that they can not be improved in general. The rates derived provide a guide to the choice of the number of simulated paths needed in optimization step, which is crucial for the good performance of any simulation based optimization algorithm. Finally, we present a numerical example of solving optimal stopping problem arising in option pricing that illustrates our theoretical findings

Spectral estimation of the fractional order of a L\'{e}vy process

We consider the problem of estimating the fractional order of a L\'{e}vy process from low frequency historical and options data. An estimation methodology is developed which allows us to treat both estimation and calibration problems in a unified way. The corresponding procedure consists of two steps: the estimation of a conditional characteristic function and the weighted least squares estimation of the fractional order in spectral domain. While the second step is identical for both calibration and estimation, the first one depends on the problem at hand. Minimax rates of convergence for the fractional order estimate are derived, the asymptotic normality is proved and a data-driven algorithm based on aggregation is proposed. The performance of the estimator in both estimation and calibration setups is illustrated by a simulation study.Comment: Published in at http://dx.doi.org/10.1214/09-AOS715 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Spectral estimation of the fractional order of a Lévy process

We consider the problem of estimating the fractional order of a L´evy process from low frequency historical and options data. An estimation methodology is developed which allows us to treat both estimation and calibration problems in a unified way. The corresponding procedure consists of two steps: the estimation of a conditional characteristic function and the weighted least squares estimation of the fractional order in spectral domain. While the second step is identical for both calibration and estimation, the first one depends on the problem at hand. Minimax rates of convergence for the fractional order estimate are derived, the asymptotic normality is proved and a data-driven algorithm based on aggregation is proposed. The performance of the estimator in both estimation and calibration setups is illustrated by a simulation study.regular Lévy processes, Blumenthal-Getoor index, semiparametric estimation

Spatial aggregation of local likelihood estimates with applications to classification

This paper presents a new method for spatially adaptive local (constant) likelihood estimation which applies to a broad class of nonparametric models, including the Gaussian, Poisson and binary response models. The main idea of the method is, given a sequence of local likelihood estimates (``weak'' estimates), to construct a new aggregated estimate whose pointwise risk is of order of the smallest risk among all ``weak'' estimates. We also propose a new approach toward selecting the parameters of the procedure by providing the prescribed behavior of the resulting estimate in the simple parametric situation. We establish a number of important theoretical results concerning the optimality of the aggregated estimate. In particular, our ``oracle'' result claims that its risk is, up to some logarithmic multiplier, equal to the smallest risk for the given family of estimates. The performance of the procedure is illustrated by application to the classification problem. A numerical study demonstrates its reasonable performance in simulated and real-life examples.Comment: Published in at http://dx.doi.org/10.1214/009053607000000271 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Statistical inference for generalized Ornstein-Uhlenbeck processes

In this paper, we consider the problem of statistical inference for generalized Ornstein-Uhlenbeck processes of the type $$X_{t} = e^{-\xi_{t}} \left( X_{0} + \int_{0}^{t} e^{\xi_{u-}} d u \right),$$ where $\xi_s$ is a L{\'e}vy process. Our primal goal is to estimate the characteristics of the L\'evy process $\xi$ from the low-frequency observations of the process $X$. We present a novel approach towards estimating the L{\'e}vy triplet of $\xi,$ which is based on the Mellin transform technique. It is shown that the resulting estimates attain optimal minimax convergence rates. The suggested algorithms are illustrated by numerical simulations.Comment: 32 pages. arXiv admin note: text overlap with arXiv:1312.473