## Convergence and stability of the balanced methods for stochastic differential equations with jumps.(English)Zbl 1236.65006

It is proved that strong balanced methods are convergent in the mean square sense with order $$1/2$$ when applied to jump diffusion stochastic differential equations of the form $dx(t)= f(x(t-))\,dt+ g(x(t-))\,dW(t)+ u(x(t-))\,dN(t),\quad t> 0,\quad x(0-)= x_0,\tag{1}$ where $$W(t)$$ is an $$m$$-dimensional Brownian motion and $$N(t)$$ is a scalar Poisson process. For sufficiently small stepsize, conditions are derived that guarantee mean-square stability of strong and weak balanced methods for equation (1). Numerical results for a few examples demonstrate the order $$1/2$$ convergence and describe superior stability of the balanced methods over the Euler-Maruyama method.

### MSC:

 65C30 Numerical solutions to stochastic differential and integral equations 65L20 Stability and convergence of numerical methods for ordinary differential equations 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 65C20 Probabilistic models, generic numerical methods in probability and statistics
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