Statistics and Probability Letters 80 (2010) 324–332 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro Successive approximation of neutral functional stochastic differential equations with jumps Brahim Boufoussi, Salah Hajji ∗ Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, 2390 Marrakesh, Morocco article info Article history: Received 9 July 2009 Received in revised form 17 September 2009 Accepted 9 November 2009 Available online 18 November 2009 MSC: 60H15 34G20 60J65 60J75 abstract By using successive approximation, we prove the existence and uniqueness result for a class of neutral functional stochastic differential equations driven both by the cylindrical Brownian motion and by the Poisson point processes in a Hilbert space with non- Lipschitzian coefficients. © 2009 Elsevier B.V. All rights reserved. 1. Introduction The purpose of this paper is to prove the existence and uniqueness of mild solutions for a class of neutral functional stochastic differential equations (FSDEs) with jumps described in the form d[x(t ) + g (t , x(t − r ))]=[Ax(t ) + f (t , x t )]dt + σ(t , x t )dW (t ) +  U h(t , x t , u) ˜ N (dt , du), 0 ≤ t ≤ T , x(t ) = ϕ(t ), −r ≤ t ≤ 0 (1) where A is the infinitesimal generator of an analytic semigroup of bounded linear operators, (T (t )) t ≥0 , in a Hilbert space H; x t ∈ D r = D([−r , 0], H) and f :[0, T ]× D r → H, g :[0, T ]× H → H,σ :[0, T ]× D r → L 2 (Q 1/2 K , H) are appropriate functions. Here L 2 (Q 1/2 K , H) denotes the space of all Q -Hilbert–Schmidt operators from Q 1/2 K into H (see Section 2). Neutral FSDEs arise in many areas of applied mathematics and such equations have received much attention in recent years. The theory of neutral FSDEs in finite-dimensional spaces has been extensively studied in the literature; see Kolmanovskii and Nosov (1986), Mao (1995, 1997), Kolmanovskii et al. (2003) and Liu and Xia (1999). However, in the infinite-dimensional Hilbert space, only a few results have been obtained in this field despite the importance and interest of the model (1). In this respect, it is worth mentioning that this kind of neutral equation arises from problems related to coupled oscillators in a noisy environment, or in problems of viscoelastic materials under random or stochastic influences (see Wu (1996) for a description of these problems in the deterministic case). To the best of our knowledge, there exist only a few published papers in this field. To be more precise, a version of (1), in the particular case where the delays are constant and h = 0, is considered in Liu (2005) and some stability properties of the mild solutions are analyzed in a similar way as in Datko (1977) in the deterministic case, while in Govindan (2005) the existence and ∗ Corresponding author. E-mail addresses: boufoussi@ucam.ac.ma (B. Boufoussi), s.hajji@ucam.ac.ma (S. Hajji). 0167-7152/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2009.11.006