ORGANIZATION IN FINANCE PREPARED BY STOCHASTIC DIFFERENTIAL EQUATIONS WITH ADDITIVE AND NONLINEAR MODELS AND CONTINUOUS OPTIMIZATION
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, the authors 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 their paper,they express them in simplified manner of approximation by both a discretization and additive models based on splines. Their parameter estimation refers to the linearly involved spline coefficients as prepared in (Taylan and Weber, 2007) and the partially nonlinearly involved probabilistic parameters. The authors 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. They also investigate when the related minimization program can be written as a Tikhonov regularization problem (sometimes called ridge regression), and they treat it using continuous optimization techniques. In particular, they 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).
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