## 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 $dX(t)= f(X(t-))\,dt+ g(X(t-))\,dW(t)+ h(X(t-))\,dN(t),\;t> 0,\;X(0-)= X_0$ 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 Numerical solutions to stochastic differential and integral equations

RODAS
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### References:

 [1] Burrage, K.; Burrage, P.M.; Tian, T., Numerical methods for strong solutions of stochastic differential equations: an overview, Proc. R. soc. London ser. A math. phys. eng. sci., 460, 373-402, (2004) · Zbl 1048.65004 [2] Chalmers, G.; Higham, D., Asymptotic stability of a jump-diffusion equation and its numerical approximation, SIAM J. sci. comp., 31, 1141-1155, (2008) · Zbl 1190.65010 [3] Gikhman, I.I.; Skorokhod, A.V., Stochastic differential equations, (1972), Springer-Verlag Berlin · Zbl 0169.48702 [4] Hairer, E.; Wanner, G., Solving ordinary differential equations II: stiff and differential – algebraic problems, (1996), Springer-Verlag Berlin · Zbl 0859.65067 [5] Higham, D., Mean-square and asymptotic stability of the stochastic theta method, SIAM J. numer. anal., 38, 753-769, (2000) · Zbl 0982.60051 [6] Higham, D., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM rev., 43, 3, 525-546, (2002) · Zbl 0979.65007 [7] Higham, D.; Kloeden, P., Numerical methods for nonlinear stochastic differential equations with jumps, Numer. math., 101, 101-119, (2005) · Zbl 1186.65010 [8] Higham, D.; Kloeden, P., Convergence and stability of implicit methods for jump-diffusion systems, Int. J. numer. anal. model., 3, 125-140, (2006) · Zbl 1109.65007 [9] Higham, D.; Mao, X.; Stuart, A.M., Exponential Mean-square stability of numerical solutions to stochastic differential equations, LMS J. comput. math., 6, 297-313, (2003) · Zbl 1055.65009 [10] Hu, Y., Semi-implicit euler – maruyama scheme for stiff stochastic equations, (), 183-202 · Zbl 0848.60057 [11] Kloeden, P.E.; Platen, E., Numerical solution of stochastic differential equations, (1992), Springer Berlin · Zbl 0925.65261 [12] Saito, Y.; Mitsui, T., Stability analysis of numerical schemes for stochastic differential equations, SIAM J. numer. anal., 33, 2254-2267, (1996) · Zbl 0869.60052 [13] Schurz, H., Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications, (1997), Logos Verlag Berlin · Zbl 0905.60002
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