Numerical regularization for SDEs: Construction of nonnegative solutions

In the numerical solution of stochastic differential equations (SDEs) such appearances as sudden, large fluctuations (explosions), negative paths or unbounded solutions are sometimes observed in contrast to the qualitative behaviour of the exact solution. To overcome this dilemma we construct regular (bounded) numerical solutions through implicit techniques without discretizing the state space. For discussion and classification, the notation of life time of numerical solutions is introduced. Thereby the task consists in construction of numerical solutions with lengthened life time up to eternal one. During the exposition we outline the role of implicitness for this "process of numerical regularization". Boundedness(Nonnegativity) of some implicit numerical solutions can be proved at least for a class of linearly bounded models. Balanced implicit methods (BIMs) turn out to be very efficient for this purpose. Furthermore, the local property of conditional positivity of numerical solutions is shown constructively (by special BIMs). The suggested approach also gives some motivation to use BIMs for the construction of numerical solutions for SDEs on bounded manifolds with "natural conditions" on their boundaries. Finally we suggest to apply these methods to population dynamics in Biology, innovation diffusion in Marketing and to mean reverting processes in Finance, such as stochastic interest rates