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Compensated stochastic theta methods for stochastic differential equations with jumps. (English) Zbl 1198.65034

Compensated stochastic theta methods (CSTM) for approximating the solutions of jumpdiffusion Ito stochastic differential equations of the form

$dX\left(t\right)=f\left(X\left(t-\right)\right)\phantom{\rule{0.166667em}{0ex}}dt+g\left(X\left(t-\right)\right)\phantom{\rule{0.166667em}{0ex}}dW\left(t\right)+h\left(X\left(t-\right)\right)\phantom{\rule{0.166667em}{0ex}}dN\left(t\right),\phantom{\rule{4pt}{0ex}}t>0,\phantom{\rule{4pt}{0ex}}X\left(0-\right)={X}_{0}$

are introduced. Mean-square convergence, A-stability, and exponential stability of CSTM methods are proved. Results of numerical experiments are presented that demonstrate a stability advantage of CSTM over stochastic theta methods.

##### MSC:
 65C30 Stochastic differential and integral equations
##### References:
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