A Note on the Solution of the Nearest Correlation Matrix Problem by Von Neumann Matrix Divergence

-- S K Mishra

In the extant literature a suggestion has been made to solve the nearest correlation matrix problem by a modified von Neumann approximation. This paper shows that obtaining the nearest positive semi-definite matrix from a given non-positivesemi- definite correlation matrix by such a method is either infeasible or suboptimal. First, if a given matrix is already positive semi-definite, there is no need to obtain any other positive semi-definite matrix closest to it. But when the given matrix is non-positive-semi-definite (Q), then a positive semi-definite matrix closest to it is needed. Then the proposed procedure fails in the absence of log(Q). But if the negative eigenvalue of Q is replaced by a zero/near-zero value, a positive semidefinite matrix is obtained, but it is not nearest to the Q matrix; there are indeed other procedures to obtain better approximation. However, the modified von Neumann approximation method yields results (although sub-optimal) and is, perhaps, one of the fastest methods most suitable to deal with larger matrices. Yet, an alternative algorithm (Fortran program available at http://www.webng.com/ economics/ncor1.txt) is provided to obtain a positive (semi-) definite matrix that performs (speed as well as accuracy-wise) much better.

© 2011 IUP. All Rights Reserved.

Efficient Simulation of the Wishart Model

-- Pierre Gauthier and Dylan Possamaï

In financial mathematics, Wishart processes have emerged as efficient tools to model stochastic covariance structures. Their numerical simulation may be quite challenging since they involve matrix processes. In this article, we propose an extensive study of financial applications of Wishart processes. First, we derive closed-form formulas for option prices in the single-asset case. Then, we show the relationship between Wishart processes and Wishart law. Finally, we review the existing discretization schemes (Euler and Ornstein-Uhlenbeck) and propose a new scheme, adapted from Heston’s QEM discretization scheme. Extensive numerical results support our comparison of these three schemes.

© 2011 IUP. All Rights Reserved.

Online PID Controller Tuning Using Fuzzy Logic Controller

-- Elijah E Omizegba, Stephen Bassi, Ejike C Anene and Usman U Abraham

In this work, a discrete Proportional-Integral-Derivative (PID) controller was trained using a Tagaki-Sugeno type fuzzy logic system. The PID parameters were acquired online without the need for manual tuning, calibration or prior knowledge of plant parameters. The developed system has the advantage of fast action and can be easily implemented with a Peripheral Interface Controller (PIC) integrated circuit. Simulation results show that the required PID gains can be acquired in less than 0.03 seconds.

© 2011 IUP. All Rights Reserved.

Numerical Solution of Run-Up Flow Through a Pipe with an Equilateral Triangular Cross-Section

-- S Raji Reddy

This paper deals with the numerical solution of the run-up flow of a viscous incompressible fluid through a pipe whose cross-section is an equilateral triangle. The problem is solved numerically, using a five-point formula. From this, it is observed that Reynold number has a great influence on the run-up flows. If it is large, the fluid comes to rest after a long time. It is also observed that on the line y = 2h, the decrease in velocity is nose symmetrical in the region. The region y > 4h is the region of silence. Hence it is concluded that Reynold numbers play an important role in run-up flows.

© 2011 IUP. All Rights Reserved.