Astability and stochastic meansquare stability
Higham, D.J. (2000) Astability and stochastic meansquare stability. BIT Numerical Mathematics, 40 (2). pp. 404409. ISSN 00063835 (http://dx.doi.org/10.1023/A:1022386822865)
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This note extends and interprets a result of Saito and Mitsui [SIAM J. Numer. Anal., 33 (1996), pp. 22542267] for a method of Milstein. The result concerns meansquare stability on a stochastic differential equation test problem with multiplicative noise. he numerical method reduces to the Theta Method on deterministic problems. Saito and Mitsui showed that the deterministic Astability property of the Theta Method does not carry through to the meansquare context in general, and gave a condition under which unconditional stability holds. The main purpose of this note is to emphasize that the approach of Saito and Mitsui makes it possible to quantify precisely the point where unconditional stability is lost in terms of the ratio of the drift (deterministic) and diffusion (stochastic) coefficients. This leads to a concept akin to deterministic A(alpha)stability that may be useful in the stability analysis of more general methods. It is also shown that meansquare Astability is recovered if the Theta Method parameter is increased beyond its normal range to the value 3/2.
ORCID iDs
Higham, D.J. ORCID: https://orcid.org/0000000266353461;

Item type: Article ID code: 178 Dates: DateEventJune 2000PublishedKeywords: meansquare, Milstein method, multiplicative noise, Theta Method, computer science, applied mathematics, Electronic computers. Computer science, Mathematics Subjects: Science > Mathematics > Electronic computers. Computer science
Science > MathematicsDepartment: Faculty of Science > Mathematics and Statistics Depositing user: Ms Sarah Scott Date deposited: 02 Mar 2006 Last modified: 18 Jun 2021 02:47 URI: https://strathprints.strath.ac.uk/id/eprint/178