Non-Lipschitz stochastic differential equations driven by multi-parameter brownian motions

Stochastics and Dynamics

Volumn 6, Issue 3, 2006, Pages 329-340

  Zhang, Xicheng ^a   Zhu, Jingyang ^a  

^a HUAZHONG UNIVERSITY OF SCIENCE AND TECHNOLOGY   (China)

Author keywords

Bihari's inequality; Discretizing approximation; Multi parameter Brownian motions; Non Lipschitz; Stochastic differential equation

Indexed keywords

EID: 33748486939     PISSN: 02194937     EISSN: None     Source Type: Journal    
DOI: 10.1142/S021949370600175X     Document Type: Article

References (20)

1. H. Airault and J. Ren, Modulus of continuity of the canonic Brownian motion "on the group of diffeomorphisms of the circle", J. Funct. Anal. 196 (2002) 395-426.
2. I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problem of differential equations, Acta Math. Acda. Sci. Hung. 7 (1956) 71-94.
3. R. Cairoli, Une inegalité pour martingale à indices multiples et ses applicatios, in Séminaire de Probabilitié, Lect. Notes in Math., Vol. 124 (Springer-Verlag, 1970), pp. 1-27.
4. R. Cairoli, Sur une équation différentielle stochastique, C. R. Acad. Sci. Paris, Ser. A 274 (1972) 1739-1742.
5. R. Cairoli and J. B. Walsh, Stochastic integrals in the plane, Acta Math. 134 (1975) 111-183.
6. G. Cao and K. He, On a type of stochastic differential equations driven by countably many Brownian motions, J. Funct. Anal. 203 (2003) 262-285.
7. Z. Huang, Stochastic integrals on general topological measurable spaces, Z. Wahsch. Gebiete 66 (1984) 25-40.
8. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North-Holland/Kodanska, 1989).
9. K. S. Lee, Y. T. Kim and J. W. Jeon, A Wong-Zakai type approximation for two-parameter processes, Stoch. Anal. Appl. 20 (2002) 615-642.
10. Z. Liang, Existence and pathwise uniqueness of solutions for stochastic differential equations with respect to the martingales in the plane, Stoch. Process. Appl. 83 (1999) 303-317.
11. J. Liu, On the existence and uniqueness of solutions to stochastic differential equations of mixed Brownian and Poisson sheet type, Stoch. Process. Appl. 94 (2001) 339-354.
12. H. W. Long, The approximation theorem of stochastic differential equations in the plane, Acta Math. Sci. 14 (1994) 272-282.
13. P. Malliavin, The canonic diffusion above the diffeomorphism group of the circle, C. R. Acad. Sci. Paris, Sér. I 329 (1999) 325-329.
14. P. Protter, Stochastic Integration and Differential Equations. A New Approach, Applications of Mathematics (New York), Vol. 21 (Springer-Verlag, 1990).
15. J. Reid, Estimate on moments of the solutions to stochastic diferential equations in the plane, Ann. Probab. 11 (1983) 656-668.
16. J. Ren and X. Zhang, Quasi-sure analysis of two-parameter stochastic differential equations, Stoch. Stoch. Rep. 72 (2002) 251-276.
17. D. Revuz and M. Yor, Continuous martingales and Brownian motion, Grund. Math. Wiss. 293 (Springer-Verlag, 1991).
18. E. Wong and M. Zakai, Martingales and stochastic integrals for processes with a multidimensional parameter, Z. Wahrsch. Gebiete 29 (1974) 109-122.
19. T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1971) 553-563.
20. J. Yeh, Estimate of strong solutions for stochastic differential equations in the plane, Pac. J. Math. 97 (1981) 217-274.