zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Convergence of numerical solution to stochastic delay differential equation with Poisson jump and Markovian switching. (English) Zbl 1120.65003
Authors’ summary: The main purpose of this paper is to study the convergence of numerical solutions to a class of stochastic delay differential equations with Poisson jump and Markovian switching. A numerical approximation scheme is proposed to approximate the solution to stochastic delay differential equations with Poisson jump and Markovian switching. It is proved that the Euler approximation solution converge to the analytic solution in probability under weaker conditions. Some known results are generalized and improved. An example is provided to illustrate our theory.

65C30Stochastic differential and integral equations
34K50Stochastic functional-differential equations
60H10Stochastic ordinary differential equations
65L20Stability and convergence of numerical methods for ODE
Full Text: DOI
[1] Svishchuk, A. V.; Kazmerchuk, Yu.I.: Stability of stochastic differential delay equations of Itô’s form with jumps and Markovian switchings, and their applications in finance. Theor. probab. Math. statist. 64, 167-178 (2002)
[2] Luo, Jiaowan: Comparison principle and stability of Itô stochastic differential delay equations with Poisson jump and Markovian switching. Nonlinear anal. 64, 253-262 (2006) · Zbl 1082.60054
[3] D.J. Higham, P.E. Kloeden, Convergence and stability of implicit methods for jump-diffusion systems, University of Strathclyde Mathematics Research Report, vol. 9, 2004. · Zbl 1109.65007
[4] Ronghua, Li; Hongbing, Meng; Yonghong, Dai: Convergence of numerical solutions to stochastic delay differential equations with jumps. Appl. math. Comput. 172, 584-602 (2006) · Zbl 1095.65006
[5] Ronghua, Li; Yingmin, Hou: Convergence and stability of numerical solutions to sddes with Markovian switching. Appl. math. Comput. 175, 1080-1091 (2006) · Zbl 1095.65005
[6] Mao, Xuerong; Matasov, Alexander; Piunovskiy, Aleksey B.: Stochastic differential delay equations with Markovian switching. Bernoulli 6, No. 1, 73-90 (2000) · Zbl 0956.60060
[7] Mao, Xuerong: Robustness of stability of stochastic differential delay equations with Markovian switching. Sacta 3, No. 1, 48-61 (2000)
[8] Shaikhet, L.: Stability of stochastic hereditary systems with Markov switching. Stochastic process. 2, No. 18, 180-184 (1996) · Zbl 0939.60049
[9] Yuan, Chenggui; Mao, Xuerong: Convergence of the Euler -- Maruyama method for stochastic differential equations with Markovian switching. Math. comput. Simulat. 64, 223-235 (2004) · Zbl 1044.65007
[10] Mao, Xuerong; Sabanis, Sotirios: Numerical solutions of stochastic differential delay equations under local Lipschitz condition. J. comput. Appl. math. 151, 215-227 (2003) · Zbl 1015.65002
[11] Mao, X.: Stochastic differential equations and applications. (1997) · Zbl 0892.60057
[12] Marion, G.; Mao, X.; Renshaw, E.: Convergence of the Euler scheme for a class of stochastic differential equations. Int. math. J. 1, 9-22 (2002) · Zbl 0987.60068