The Weak Convergence Rate of Two Semi-Exact Discretization Schemes for the Heston Model

Inspired by the article Weak Convergence Rate of a Time-Discrete Scheme for the Heston Stochastic Volatility Model, Chao Zheng, SIAM Journal on Numerical Analysis 2017, 55:3, 1243–1263, we studied the weak error of discretization schemes for the Heston model, which are based on exact simulation of the underlying volatility process. Both for an Euler- and a trapezoidal-type scheme for the log-asset price, we established weak order one for smooth payoffs without any assumptions on the Feller index of the volatility process. In our analysis, we also observed the usual trade off between the smoothness assumption on the payoff and the restriction on the Feller index. Moreover, we provided error expansions, which could be used to construct second order schemes via extrapolation. In this paper, we illustrate our theoretical findings by several numerical examples.

Levenberg-Marquardt Neural Network for Eye States Detection Based on Electroencephalography Data

The eye state detection is one of various task toward Brain Computer Interface system. The eye state can be read in brain signals. In this paper use EEG Eye State dataset (Rosler, 2013) from UCI Machine Learning Repository Database. Dataset is consisting of continuous 14 EEG measurements in 117 seconds. The eye states were marked as “1” or “0”. “1” indicates the eye-closed and “0” the eye-open state. The proposed schemes use Multi Layer Neural Network with Levenberg Marquardt optimization learning algorithm, as classification method. Levenberg Marquardt method used to optimize the learning algorithm of neural network, because the standard algorithm has a weak convergence rate. It is need many iterations to have minimum error. Based on the analysis towards the experiment on the EEG dataset, it can be concluded that the proposed scheme can be implemented to detect the Eye State. The best accuracy gained from combination variable sigmoid function, data normalization and number of neurons are 31 (95.71%) for one hidden layer, and 98.912% for two hidden layers with number of neurons are 39 and 47 neurons and linear function.