Approximation of Stochastic Differential Equations by Additive Models Using Splines and Conic Programming

Stochastic differential equations are widely used to model noise-affected phenomena in nature, technology and economy (Kloeden et al., 1994). As these equations are usually hard to represent by a computer and hard to resolve we express them in simplified manner. We introduce an approximation by discretization and additive models based on splines. Then, we construct a penalized residual sum of squares (PRSS) for this model. We show when the related minimization program can be written as a Tikhonov regularization problem (ridge regression), and we treat it using continuous optimization techniques. In particular, we apply the elegant framework of conic quadratic programming. Convex optimization problems are very well-structured, resembling linear programs and permit the use of interior point methods (Nesterov & Nemirovskii, 1993).