zbMATH — the first resource for mathematics

Successive approximation of neutral functional stochastic differential equations with jumps. (English) Zbl 1196.60114
Existence and uniquenss of càdlàg mild solutions of a stochastic delay equation \[ d[x(t)+g(t,x(t-r))]=[Ax(t)+f(t,x_t)]\,dt+\sigma(t,x_t)\,dW(t)+\int_{\mathcal U}h(t,x_t,u)\tilde N(dt,du) \] in a Hilbert space \(H\) with an initial condition \(x(t)=\varphi(t)\) for \(t\in[-r,0]\) is proved. Here \(x_t(s)=x(t+s)\), \(s\in[-r,0]\), \(A\) generates a holomorphic semigroup of contractions on \(H\), \(W\) is a cylindrical Wiener process, \(\tilde N\) is a compensated Poisson martingale measure generated by a stationary Poisson point process in a \(\sigma\)-finite measure space \((\mathcal U,\mathcal E,\nu)\), the nonlinearities \(g\), \(f\), \(\sigma\) and \(h\) are defined on suitable spaces and, roughly speaking, \(g\) is Lipschitz of at most linear growth and the modulus of continuity of \(f\), \(\sigma\) and \(h\) is at most \(\varepsilon\max\{1,|\rho(\varepsilon)|\}\) where \(\rho\) is of multiples of iterated logarithms growth near the origin.

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
34G20 Nonlinear differential equations in abstract spaces
60J65 Brownian motion
60J75 Jump processes (MSC2010)
Full Text: DOI
[1] Albeverio, S.; Mandrekar, V.; Rüdiger, B., Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise, Stochastic process. appl., 119, 835-863, (2009) · Zbl 1168.60014
[2] Bihari, I., A generalization of a lemma of belmman and its application to uniqueness problem of differential equations, Acta. math., acad. sci. hungar., 7, 71-94, (1956) · Zbl 0070.08201
[3] Caraballo, T.; Real, J.; Taniguchi, T., The exponential stability of neutral stochastic delay partial differential equations, Discrete contin. dyn. syst., 18, 295-313, (2007) · Zbl 1125.60059
[4] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press Cambridge · Zbl 0761.60052
[5] Datko, R., Linear autonomous neutral differential equations in Banach spaces, J. differential equations, 25, 258-274, (1977) · Zbl 0402.34066
[6] Goldstein, A.; Jerome, Semigroups of linear operators and applications, () · Zbl 1364.47005
[7] Govindan, T.E., Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics, 77, 139-154, (2005) · Zbl 1115.60064
[8] Hausenblas, E; Seidler, J., Stochastic convolutions driven by martingales: maximal inequalities and exponential integrability, Stoch. anal. appl., 26, 1, 98-119, (2008) · Zbl 1153.60035
[9] Ikeda, N.; Watanabe, S., Stochastic differential equations and diffusion processes, (1989), Nort-Holland, Kodansha Amsterdam, Oxford, New York · Zbl 0684.60040
[10] Kolmanovskii, V.; Koroleva, N.; Maizenberg, T.; Mao, X.; Matasov, A., Neutral stochastic differential delay equations with Markovian switching, Stoch. anal. appl., 21, 4, 819-847, (2003) · Zbl 1025.60028
[11] Kolmanovskii, V.B.; Nosov, V.R., Stability of functional differential equations, (1986), Academic Press · Zbl 0593.34070
[12] Liu, K., Uniform stability of autonomous linear stochastic fuctional diferential equations in infinite dimensions, Stochastic process. appl., 115, 1131-1165, (2005) · Zbl 1075.60078
[13] Liu, K.; Xia, X., On the exponential stability in Mean square of neutral stochastic functional differential equations, Systems control lett., 37, 4, 207-215, (1999) · Zbl 0948.93060
[14] Mahmudov, N.I., Existence and uniqueness results for neutral SDEs in Hilbert spaces, Stochastic analysis and applications, 24, 79-95, (2006) · Zbl 1110.60063
[15] Mao, X., Exponential stability in Mean square of neutral stochastic differential functional equations, Systems control lett., 26, 245-251, (1995) · Zbl 0877.93133
[16] Mao, X., Razumikhin-type theorems on exponential stability of neutral stochastic functional – differential equations, SIAM J. math. anal., 28, 2, 389-401, (1997) · Zbl 0876.60047
[17] Pazy, A., ()
[18] Wu, J., ()
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.