Fu, Zongfei; Li, Zenghu Stochastic equations of non-negative processes with jumps. (English) Zbl 1184.60022 Stochastic Processes Appl. 120, No. 3, 306-330 (2010). Summary: We study stochastic equations of non-negative processes with jumps. The existence and uniqueness of strong solutions are established under Lipschitz and non-Lipschitz conditions. Under suitable conditions, the comparison properties of solutions are proved. Those results are applied to construct continuous state branching processes with immigration as strong solutions of stochastic equations. Cited in 2 ReviewsCited in 87 Documents MSC: 60H20 Stochastic integral equations 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) Keywords:stochastic equation; strong solution; pathwise uniqueness; comparison theorem; non-Lipschitz condition; continuous state branching process; immigration × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Aldous, D., Stopping times and tightness, Ann. Probab., 6, 335-340 (1978) · Zbl 0391.60007 [2] Bass, R. F., (Stochastic differential equations driven by symmetric stable processes. Stochastic differential equations driven by symmetric stable processes, Lecture Notes Math., vol. 1801 (2003), Springer-Verlag: Springer-Verlag Berlin), 302-313 · Zbl 1039.60056 [3] Bass, R. F., Stochastic differential equations with jumps, Probab. Surv., 1, 1-19 (2004) · Zbl 1189.60114 [4] Bass, R. F.; Burdzy, K.; Chen, Z.-Q., Stochastic differential equations driven by stable processes for which pathwise uniqueness fails, Stochastic Process. Appl., 111, 1-15 (2004) · Zbl 1111.60038 [5] Bertoin, J.; Le Gall, J.-F., Stochastic flows associated to coalescent processes III: Limit theorems, Illinois J. Math., 50, 147-181 (2006) · Zbl 1110.60026 [6] Cherny, A. S.; Engelbert, H.-J., (Singular Stochastic Differential Equations. Singular Stochastic Differential Equations, Lecture Notes Math., vol. 1858 (2005), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1071.60003 [7] Ph. Courrége, Sur la forme intégro-différentielle des opérateurs de \(C_K^\inftyC \); Ph. Courrége, Sur la forme intégro-différentielle des opérateurs de \(C_K^\inftyC \) [8] Dawson, D. A.; Li, Z. H., Skew convolution semigroups and affine Markov processes, Ann. Probab., 34, 1103-1142 (2006) · Zbl 1102.60065 [9] Dellacherie, C.; Meyer, P. A., Probabilites and Potential (1982), North-Holland: North-Holland Amsterdam, (Chapters V-VIII) · Zbl 0494.60002 [10] Duffie, D.; Filipović, D.; Schachermayer, W., Affine processes and applications in finance, Ann. Appl. Probab., 13, 984-1053 (2003) · Zbl 1048.60059 [11] Ethier, S. N.; Kurtz, T. G., Markov Processes: Characterization and Convergence (1986), Wiley: Wiley New York · Zbl 0592.60049 [12] Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes (1989), North-Holland/Kodasha: North-Holland/Kodasha Amsterdam/Tokyo · Zbl 0684.60040 [13] Kawazu, K.; Watanabe, S., Branching processes with immigration and related limit theorems, Theory Probab. Appl., 16, 36-54 (1971) · Zbl 0242.60034 [14] Lambert, A., Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct, Electron. J. Probab., 12, 420-446 (2007) · Zbl 1127.60082 [15] Lamberton, D.; Lapeyre, B., Introduction to Stochastic Calculus Applied to Finance (1996), Chapman and Hall: Chapman and Hall London · Zbl 0898.60002 [16] Protter, P. E., Stochastic Integration and Differential Equations (2003), Springer-Verlag: Springer-Verlag Berlin [17] Rozovsky, B. L., A note on strong solutions of stochastic differential equations with random coefficients, (Stochastic Differential Systems. Stochastic Differential Systems, Proc. IFIP-WG 7/1 Working Conf., Vilnius, 1978. Stochastic Differential Systems. Stochastic Differential Systems, Proc. IFIP-WG 7/1 Working Conf., Vilnius, 1978, Lecture Notes Control Inform. Sci., vol. 25 (1980), Springer-Verlag: Springer-Verlag Berlin), 287-296 · Zbl 0481.60063 [18] Sato, K., Lévy Processes and Infinitely Divisible Distributions (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0973.60001 [19] Stroock, D. W., Diffusion processes associated with Lévy generators, Z. Wahrsch. verw. Geb., 32, 209-244 (1975) · Zbl 0292.60122 [20] Yamada, T.; Watanabe, S., On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 11, 155-167 (1971) · Zbl 0236.60037 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.