# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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

$d\left[x\left(t\right)+g\left(t,x\left(t-r\right)\right)\right]=\left[Ax\left(t\right)+f\left(t,{x}_{t}\right)\right]\phantom{\rule{0.166667em}{0ex}}dt+\sigma \left(t,{x}_{t}\right)\phantom{\rule{0.166667em}{0ex}}dW\left(t\right)+{\int }_{𝒰}h\left(t,{x}_{t},u\right)\stackrel{˜}{N}\left(dt,du\right)$

in a Hilbert space $H$ with an initial condition $x\left(t\right)=\varphi \left(t\right)$ for $t\in \left[-r,0\right]$ is proved. Here ${x}_{t}\left(s\right)=x\left(t+s\right)$, $s\in \left[-r,0\right]$, $A$ generates a holomorphic semigroup of contractions on $H$, $W$ is a cylindrical Wiener process, $\stackrel{˜}{N}$ is a compensated Poisson martingale measure generated by a stationary Poisson point process in a $\sigma$-finite measure space $\left(𝒰,ℰ,\nu \right)$, 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 $\epsilon max\left\{1,|\rho \left(\epsilon \right)|\right\}$ where $\rho$ is of multiples of iterated logarithms growth near the origin.

##### MSC:
 60H15 Stochastic partial differential equations 34G20 Nonlinear ODE in abstract spaces 60J65 Brownian motion 60J75 Jump processes