TY - JOUR
T1 - An Axiomatic Approach to Numerical Approximations of Stochastic Processes
AU - Schurz , Henri
JO - International Journal of Numerical Analysis and Modeling
VL - 4
SP - 459
EP - 480
PY - 2006
DA - 2006/03
SN - 3
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/913.html
KW - stochastic differential equations, numerical methods, stochastic difference equations, convergence, stability, contractivity, stochastic Kantorovich-Lax-Richtmeyer principle, Lyapunov-type functions, worst case convergence rates.
AB - <p style="text-align: justify;">An axiomatic approach to the numerical approximation $Y$ of
some stochastic process $X$ with values on a separable Hilbert space $H$ is
presented by means of Lyapunov-type control functions $V$. The processes $X$ and $Y$ are interpreted as flows of stochastic differential and difference
equations, respectively. The main result is the proof of some extensions
of well-known deterministic principle of Kantorovich-Lax-Richtmeyer to
approximate solutions of initial value differential problems to the
stochastic case. The concepts of invariance, smoothness of martingale
parts, consistency, stability, and contractivity of stochastic processes
are uniquely combined to derive efficient convergence rates on finite
and infinite time-intervals. The applicability of our results is
explained with drift-implicit backward Euler methods applied to ordinary
stochastic differential equations (SDEs) driven by standard Wiener
processes on Euclidean spaces $H = \mathbb{R}^d$ along functions such as $V(x) = \sum_{i=0} ^k c_i x^{2i}$. A detailed discussion on an example with cubic
nonlinearity from field theory in physics (stochastic Ginzburg-Landau
equation) illustrates the suggested axiomatic approach.</p>