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