# 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)
Convergence rate of numerical solutions to SFDEs with jumps. (English) Zbl 1236.65005

The authors consider stochastic functional differential equations (SFDEs) with jumps of the form

$dx\left(t\right)=f\left({x}_{t}\right)dt+g\left({x}_{t}\right)d{B}_{t}+h\left({x}_{t}\right)dN\left(t\right),\phantom{\rule{4pt}{0ex}}0\le t\le T,$

with given ${x}_{0}$, where $x$ is $n$-dimensional,$\phantom{\rule{4pt}{0ex}}{x}_{t}:=\left\{x\left(t+\theta \right),\phantom{\rule{4pt}{0ex}}-\tau \le \theta \le 0\right\}$, ${x}_{{t}^{-}}:=\left\{x\left({\left(t+\theta \right)}^{-}\right),\phantom{\rule{4pt}{0ex}}-\tau \le \theta \le 0\right\}$,$\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}x\left({t}^{-}\right):={lim}_{s↑t}x\left(s\right)$, $B\left(t\right)$ is an $m$-dimensional Brownian motion, and $N\left(t\right)$ is a scalar Poisson process.

Under a global Lipschitz condition they show that the $p$th-moment convergence of Euler-Maruyama numerical solutions to SFDEs with jumps has the order $1/p$ for any $p\ge 2$. This is different from the case of SFDEs without jumps, where the order is $1/2$ for any $p\ge 2$. They consider also the mean-square convergence under a local Lipschitz condition.

##### MSC:
 65C30 Stochastic differential and integral equations 65L20 Stability and convergence of numerical methods for ODE 60H35 Computational methods for stochastic equations 34K50 Stochastic functional-differential equations 34K28 Numerical approximation of solutions of functional-differential equations 34F05 ODE with randomness 60H10 Stochastic ordinary differential equations 60J65 Brownian motion