Organization in Finance Prepared by Stochastic Differential Equations with Additive and Nonlinear Models and Continuous Optimization

Taylan Pakize and Weber Gerhard-Wilhelm
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Taylan Pakize: Institute of Applied Mathematics, Middle East Technical University, 06531 Ankara, Turkey Department of Mathematics, Dicle University, 21280 Diyarbakir, Turkey
Weber Gerhard-Wilhelm: Institute of Applied Mathematics, Middle East Technical University, 06531 Ankara, Turkey Faculty of Economics, Management and Law, University of Siegen, 57076 Siegen, Germany

Organizacija, 2008, vol. 41, issue 5, 185-193

Abstract: A central element in organization of financal means by a person, a company or societal group consists in the constitution, analysis and optimization of portfolios. This requests the time-depending modeling of processes. Likewise many processes in nature, technology and economy, financial processes suffer from stochastic fluctuations. Therefore, we consider stochastic differential equations (Kloeden, Platen and Schurz, 1994) since in reality, especially, in the financial sector, many processes are affected with noise. As a drawback, these equations are hard to represent by a computer and hard to resolve. In our paper, we express them in simplified manner of approximation by both a discretization and additive models based on splines. Our parameter estimation refers to the linearly involved spline coefficients as prepared in (Taylan and Weber, 2007) and the partially nonlinearly involved probabilistic parameters. We construct a penalized residual sum of square for this model and face occuring nonlinearities by Gauss-Newton's and Levenberg-Marquardt's method on determining the iteration step. We also investigate when the related minimization program can be written as a Tikhonov regularization problem (sometimes called ridge regression), and we treat it using continuous optimization techniques. In particular, we prepare access to the elegant framework of conic quadratic programming. These convex optimation problems are very well-structured, herewith resembling linear programs and, hence, permitting the use of interior point methods (Nesterov and Nemirovskii, 1993).

Keywords: Stochastic Differential Equations; Regression; Statistical Learning; Parameter Estimation; Splines; Gauss-Newton Method; Levenberg-Marquardt's method; Smoothing; Stability; Penalty Methods; Tikhonov Regularization; Continuous Optimization; Conic Quadratic Programming; Stochastic Differential Equations; Regression; Statistical Learning; Parameter Estimation; Splines; Gauss-Newton Method; Levenberg-Marquardt's method; Smoothing; Stability; Penalty Methods; Tikhonov Regularization; Continuous Optimization; Conic Quadratic Programming (search for similar items in EconPapers)
Date: 2008
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Persistent link: https://EconPapers.repec.org/RePEc:vrs:organi:v:41:y:2008:i:5:p:185-193:n:3

DOI: 10.2478/v10051-008-0020-8

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