Volume 3, Issue 4
An Axiomatic Approach to Numerical Approximations of Stochastic Processes

Henri Schurz

Int. J. Numer. Anal. Mod., 3 (2006), pp. 459-480.

Published online: 2006-03

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  • Abstract

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.

  • Keywords

stochastic differential equations, numerical methods, stochastic difference equations, convergence, stability, contractivity, stochastic Kantorovich-Lax-Richtmeyer principle, Lyapunov-type functions, worst case convergence rates.

  • AMS Subject Headings

65C20, 65C30, 65L20, 65D30, 34F05, 37H10, 60H10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-3-459, author = {Schurz , Henri}, title = {An Axiomatic Approach to Numerical Approximations of Stochastic Processes}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2006}, volume = {3}, number = {4}, pages = {459--480}, abstract = {

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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/913.html} }
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 -

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.

Henri Schurz. (1970). An Axiomatic Approach to Numerical Approximations of Stochastic Processes. International Journal of Numerical Analysis and Modeling. 3 (4). 459-480. doi:
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