A brief review on stability investigations of numerical methods for systems of stochastic differential equations

Henri Schurz; Henri Schurz

doi:10.3934/nhm.2024016

Networks and Heterogeneous Media

2024, Volume 19, Issue 1: 355-383. doi: 10.3934/nhm.2024016

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A brief review on stability investigations of numerical methods for systems of stochastic differential equations

Henri Schurz ^,

Department of Mathematics, Southern Illinois University, 1245 Lincoln Drive, Carbondale, IL 62901, USA

Received: 20 January 2024 Revised: 13 March 2024 Accepted: 21 March 2024 Published: 01 April 2024

A modest review on stability investigations of numerical methods for systems of Itô-interpreted stochastic differential equations (SDEs) driven by $ m $-dimensional Wiener processes $ W = (W^1, W^2, ..., W^m) $ is presented in $ \mathbb{R}^d $. Since the problem of relevance of 1D test equations for multidimensional numerical methods has not completely been solved so far, we suggest to use the Krein-Perron-Frobenius theory of positive operators on positive cones of $ \mathbb{R}^{d \times d} $, instead of classic stability functions with values in $ \mathbb{C}^1 $, which is only relevant for the very restricted case of "simultaneously diagonalizable" SDEs. Our main focus is put on the concept of asymptotic mean square and almost sure (a.s.) stability for systems with state-dependent noise (multiplicative case), and the concept of exact preservation of asymptotic probabilistic quantities for systems with state-independent noise (additive case). The asymptotic exactness of midpoint methods with any equidistant step size $ h $ is worked out in order to underline their superiority within the class of all drift-implicit, classic theta methods for multidimensional, bilinear systems of SDEs. Balanced implicit methods with appropriate weights can also provide a.s. exact, asymptotically stable numerical methods for pure diffusions. The review on numerical stability is based on "major breakthroughs" of research for systems of SDEs over the last 35 years, with emphasis on applicability to all dimensions $ d \ge 1 $.
- stochastic differential equations,
- numerical methods,
- almost sure stability,
- mean square stability,
- moment stability,
- test equations,
- stochastic theta methods,
- theta Milstein methods
Citation: Henri Schurz. A brief review on stability investigations of numerical methods for systems of stochastic differential equations[J]. Networks and Heterogeneous Media, 2024, 19(1): 355-383. doi: 10.3934/nhm.2024016

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Abstract

A modest review on stability investigations of numerical methods for systems of Itô-interpreted stochastic differential equations (SDEs) driven by $ m $-dimensional Wiener processes $ W = (W^1, W^2, ..., W^m) $ is presented in $ \mathbb{R}^d $. Since the problem of relevance of 1D test equations for multidimensional numerical methods has not completely been solved so far, we suggest to use the Krein-Perron-Frobenius theory of positive operators on positive cones of $ \mathbb{R}^{d \times d} $, instead of classic stability functions with values in $ \mathbb{C}^1 $, which is only relevant for the very restricted case of "simultaneously diagonalizable" SDEs. Our main focus is put on the concept of asymptotic mean square and almost sure (a.s.) stability for systems with state-dependent noise (multiplicative case), and the concept of exact preservation of asymptotic probabilistic quantities for systems with state-independent noise (additive case). The asymptotic exactness of midpoint methods with any equidistant step size $ h $ is worked out in order to underline their superiority within the class of all drift-implicit, classic theta methods for multidimensional, bilinear systems of SDEs. Balanced implicit methods with appropriate weights can also provide a.s. exact, asymptotically stable numerical methods for pure diffusions. The review on numerical stability is based on "major breakthroughs" of research for systems of SDEs over the last 35 years, with emphasis on applicability to all dimensions $ d \ge 1 $.

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