Estimates on moments of the solutions to stochastic differential equations with respect to martingales in the plane

Let M = { M z , z ϵ R 2 +} be a two-parameter strong martingale, A be a two-parameter increasing process on R 2 + = [0, + ∞) × [0, + ∞). Consider the following stochastic differential equations in the plane: X z = X 0 + ∞ R z a(ξ,X) dM ξ + ∞ R z b(ξ,X) dA ξ for z ϵ R 2 +. Under some assumptions on t...

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Published inStochastic processes and their applications Vol. 62; no. 2; pp. 263 - 276
Main Authors Liang, Zong-xia, Zheng, Ming-li
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.07.1996
Elsevier
SeriesStochastic Processes and their Applications
Subjects
60H15
60H15 60H20 Two-parameter stochastic differential equation Two-parameter strong martingale Two-parameter Ito's formula Gronwall's inequality
60H20
Gronwall's inequality
Two-parameter Ito's formula
Two-parameter stochastic differential equation
Two-parameter strong martingale
60H20
Two-parameter Ito's formula
Two-parameter stochastic differential equation
60H15
Gronwall's inequality
Two-parameter strong martingale
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Abstract Let M = { M z , z ϵ R 2 +} be a two-parameter strong martingale, A be a two-parameter increasing process on R 2 + = [0, + ∞) × [0, + ∞). Consider the following stochastic differential equations in the plane: X z = X 0 + ∞ R z a(ξ,X) dM ξ + ∞ R z b(ξ,X) dA ξ for z ϵ R 2 +. Under some assumptions on the coefficients a, b and the integrators M, A, we prove the existence and uniqueness of solutions for the equations, and obtain some estimates on moments of solution.
AbstractList Let M = {Mz, z [epsilon] R2+} be a two-parameter strong martingale, A be a two-parameter increasing process on R2+ = [0, + [infinity]) x [0, + [infinity]). Consider the following stochastic differential equations in the plane: for z [epsilon] R2+. Under some assumptions on the coefficients a, b and the integrators M, A, we prove the existence and uniqueness of solutions for the equations, and obtain some estimates on moments of solution.
Let M = { M z , z ϵ R 2 +} be a two-parameter strong martingale, A be a two-parameter increasing process on R 2 + = [0, + ∞) × [0, + ∞). Consider the following stochastic differential equations in the plane: X z = X 0 + ∞ R z a(ξ,X) dM ξ + ∞ R z b(ξ,X) dA ξ for z ϵ R 2 +. Under some assumptions on the coefficients a, b and the integrators M, A, we prove the existence and uniqueness of solutions for the equations, and obtain some estimates on moments of solution.
Author Liang, Zong-xia
Zheng, Ming-li
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Cites_doi 10.1016/0304-4149(78)90031-5
10.1214/aop/1176996028
10.2140/pjm.1981.97.217
10.1007/BF02392100
10.1214/aop/1176995718
10.1214/aop/1176993510
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Keywords 60H20
Two-parameter Ito's formula
Two-parameter stochastic differential equation
60H15
Gronwall's inequality
Two-parameter strong martingale
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Chevalier (BIB2) 1982; 106
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Wong, Zakai (BIB9) 1978; 6
Yeh (BIB10) 1981; 97
Ikeda, Watanabe (BIB3) 1981
Cairoli, Walsh (BIB1) 1975; 134
Wong, Zakai (BIB7) 1976; 4
Yeh (10.1016/0304-4149(96)00057-9_BIB10) 1981; 97
Wong (10.1016/0304-4149(96)00057-9_BIB7) 1976; 4
Merzbach (10.1016/0304-4149(96)00057-9_BIB5) 1980; 53
Wong (10.1016/0304-4149(96)00057-9_BIB8) 1977; 5
Chevalier (10.1016/0304-4149(96)00057-9_BIB2) 1982; 106
Reid (10.1016/0304-4149(96)00057-9_BIB6) 1983; 11
Ikeda (10.1016/0304-4149(96)00057-9_BIB3) 1981
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Nie (10.1016/0304-4149(96)00057-9_BIB4) 1987; 2
Wong (10.1016/0304-4149(96)00057-9_BIB9) 1978; 6
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Snippet Let M = { M z , z ϵ R 2 +} be a two-parameter strong martingale, A be a two-parameter increasing process on R 2 + = [0, + ∞) × [0, + ∞). Consider the following...
Let M = {Mz, z [epsilon] R2+} be a two-parameter strong martingale, A be a two-parameter increasing process on R2+ = [0, + [infinity]) x [0, + [infinity])....
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SubjectTerms 60H15
60H15 60H20 Two-parameter stochastic differential equation Two-parameter strong martingale Two-parameter Ito's formula Gronwall's inequality
60H20
Gronwall's inequality
Two-parameter Ito's formula
Two-parameter stochastic differential equation
Two-parameter strong martingale
Title Estimates on moments of the solutions to stochastic differential equations with respect to martingales in the plane
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