zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Almost sure exponential stability of numerical solutions for stochastic delay differential equations. (English) Zbl 1193.65009

A theorem is proved that gives sufficient conditions for almost sure exponential stability (ASES) of Euler-Maruyama method numerical solutions of the n-dimensional nonlinear stochastic delay differential equation

dx(t)=f(x(t),x(t-τ),t)dt+g(x(t),x(t-τ),t)dw(t),t0·

A counterexample is presented to show that without the linear growth condition on f of the theorem, ASES may be lost. Then for the backward Euler-Maruyama method ASES is proved when a one-sided Lipschitz condition on f in x replaces the linear growth condition on f.


MSC:
65C30Stochastic differential and integral equations
65C99Probabilistic methods, simulation and stochastic differential equations (numerical analysis)
Software:
RODAS
References:
[1]Baker C.T.H., Buckwar E.: Numerical analysis of explicit one-step methods for stochastic delay differential equations. LMS J. Comput. Math. 3, 315–335 (2000) · Zbl 0974.65008 · doi:10.1112/S1461157000000322
[2]Baker C.T.H., Buckwar E.: Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations. J. Comput. Appl. Math. 184, 404–427 (2005) · Zbl 1081.65011 · doi:10.1016/j.cam.2005.01.018
[3]Burrage K., Burrage P., Mitsui T.: Numerical solutions of stochastic differential equations–implematation and stability issues. J. Comput. Appl. Math. 125, 171–182 (2000) · Zbl 0971.65003 · doi:10.1016/S0377-0427(00)00467-2
[4]Burrage K., Tian T.: A note on the stability propertis of the Euler methods for solving stochastic differential equations. N Z J. Math. 29, 115–127 (2000)
[5]Hairer E., Wanner G.: Solving Ordinary Differential Equation II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (1996)
[6]Higham D.J.: Mean-square and asymptotic stability of the stochastic theta methods. SIAM J. Numer. Anal. 38, 753–769 (2000) · Zbl 0982.60051 · doi:10.1137/S003614299834736X
[7]Higham D.J., Mao X., Yuan C.: Almost sure and Moment exponential stability in the numerical simulation of stochastic differential equations. SIAM J. Numer. Anal. 45, 592–607 (2007) · Zbl 1144.65005 · doi:10.1137/060658138
[8]Kloeden P.E., Platen E.: The Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)
[9]Liptser R.Sh., Shiryaev A.N.: Theory of Martingale. Kluwer Academic Publishers, Dordrecht (1989)
[10]Mao X.: Approximate solutions for a class of stochastic evolution equations with variable delays–part II. Numer. Funct. Anal. Optim. 15, 65–76 (1994) · Zbl 0796.60068 · doi:10.1080/01630569408816550
[11]Mao X.: Exponential Stability of Stochastic Differential Equation. Marcel Dekker, New York (1994)
[12]Mao X.: Stochastic Differential Equations and their Applications. Horwood, Chichester (1997)
[13]Mao X.: Stochastic versions of the LaSalle theorem. J. Differ. Equ. 153, 175–195 (1999) · Zbl 0921.34057 · doi:10.1006/jdeq.1998.3552
[14]Mao X.: LaSalle-type theorems for stochastic differential delay equations. J. Math. Anal. Appl. 236, 350–369 (1999) · Zbl 0958.60057 · doi:10.1006/jmaa.1999.6435
[15]Mao X.: The LaSalle-type theorems for stochastic differential equations. Nonlinear Stud. 7, 307–328 (2000)
[16]Mao X.: A note on the LaSalle-type theorems for stochastic differential delay equations. J. Math. Anal. Appl. 268, 125–142 (2002) · Zbl 0996.60064 · doi:10.1006/jmaa.2001.7803
[17]Mao X.: Numerical solutions of stochastic functional differential equations. LMS J. Comput. Math. 6, 141–161 (2003)
[18]Mao X.: Exponential stability of equidistant Euler-Maruyama approximations of stochastic differential delay equations. J. Comput. Appl. Math. 200, 297–316 (2007) · Zbl 1114.65005 · doi:10.1016/j.cam.2005.11.035
[19]Mao X., Rassias M.J.: Khasminskii-type theorems for stochastic differential delay equations. Stoch. Anal. Appl. 23, 1045–1069 (2005) · Zbl 1082.60055 · doi:10.1080/07362990500118637
[20]Mao X., Yuan C.: Stochastic Differential Equations with Markovian Switching. Imperial College Press, London (2006)
[21]Pang S., Deng F., Mao X.: Almost sure and moment exponential stability of Euler–Maruyama discretizations for hybrid stochastic differential equations. J. Comput. Appl. Math. 213, 127–141 (2008) · Zbl 1141.65006 · doi:10.1016/j.cam.2007.01.003
[22]Rodkina A., Basin M.: On delay-dependent stability for vector nonlinear stochastic delay-difference equations with Volterra diffusion term. Syst. Control Lett. 56, 423–430 (2007) · Zbl 1124.93066 · doi:10.1016/j.sysconle.2006.11.001
[23]Rodkina A., Schurz H.: Almost sure asymptotic stability of drift-implicit θ-methods for bilinear ordinary stochastic differential equations in 1 . J. Comput. Appl. Math. 180, 13–31 (2005) · Zbl 1073.65009 · doi:10.1016/j.cam.2004.09.060
[24]Saito Y., Mitsui T.: T-stability of numerical scheme for stochastic differential equations. World Sci. Ser. Appl. Anal. 2, 333–344 (1993)
[25]Saito Y., Mitsui T.: Stability analysis of numerical schemes for stochastic differential equations. SIAM J. Numer. Anal. 33, 2254–2267 (1996) · Zbl 0869.60052 · doi:10.1137/S0036142992228409
[26]Shiryaev A.N.: Probability. Springer, Berlin (1996)