Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications

Henri Schurz

ISBN 978-3-931216-94-8
264 pages, year of publication: 1997
price: 45.00 €

Stochastic differential equations play an important role in almost all fields of Natural Sciences, Engineering, Mathematical Finance and Marketing. Usually, one can not solve them analytically. Therefore one has to resort to numerical techniques. The analysis of corresponding algorithms with respect to convergence rates is well-understood nowadays. In contrast to that, there is a lack of systematic stability analysis (cf. famous Lax-Theorem in Numerical Analysis). The presented monograph tries to close this gap by rigorous investigation of stability properties of the simplest stochastic-numerical methods as parameterized families of implicit Euler, Milstein and Balanced methods for these equations on a multidimensional level (since, in general, there is no counterpart to the deterministic concept of test equations in Stochastics so far). Here we work out the outstanding role of implicit trapezoidal (midpoint) rule for adequate detection of stationary probabilistic laws of underlying continuous time systems. Furthermore, Balanced implicit methods can be appropriate for construction of "numerical solutions" on bounded submanifolds without loosing convergence speeds and without discretization underlying state spaces under some mild assumptions. The stability analysis is carried out for linear (divided into that for systems with multiplicative and that with additive noise) as well as nonlinear dissipative stochastic systems using theory of positive operators and monotonocity properties.

Besides, we introduce classes of linear-implicit numerical methods and prove their convergence rates. The presented analysis can be carried over to other numerical methods, and hence delivers a tool to investigate further, more complicated methods such as those of higher order of convergence in a very systematic way. Eventually, the results are tested and graphically illustrated by a series of applications, e.g. for Stochastic Kubo Oscillator, Planar Brusselator, Interest Rates, Brownian Bridges, Innovation Processes, Oscillating Systems under Random Seismic Vibrations with Time-Delays, Duffing and Van der Pol Oscillators.

The book ends with an almost exhaustive list on reviewed literature related to this subject, sorted into theory and numerics.