Stochastic Differential Equations

Random effects can be introduced in dynamical systems. For example, the aim is to include noise in the mathematical model. In case of time-dependent ordinary differential equations, the noise is added in form of the increments of a Wiener process. It follows a system of stochastic differential equations (SDEs). The definition of the solution of SDEs is based on stochastic integrals, where mainly the two concepts of Ito and Stratonovich are applied. Consequently, the analysis as well as the numerical solution of SDEs becomes much more involved in comparison to ordinary differential equations. Efficient numerical methods are required in practice. Generalisations of the SDEs based on Wiener processes exist in form of jump processes and others. The field of computational finance applies these models, where asset prices are described by SDEs, for example. Furthermore, climate models sometimes use SDEs to simulate random changes in the temperature.