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Simulation and approximation of Lévy-driven stochastic differential equations. (English) Zbl 1273.60080

The author considers the numerical solution of a scalar stochastic differential equation \[ X_t=x+\int_0^t \sigma(X_{s-})dZ_s \] driven by a one-dimensional square-integrable Lévy process \(Z\). The numerical method employed is an approximate Euler scheme where the small jumps of the Lévy process are approximated by Gaussian variables. The first main result of the paper is that in this case the strong error is smaller than previously proven. This result is proven for bounded and Lipschitz continuous coefficients \(\sigma\) but afterwards generalized to locally Lipschitz coefficients satisfying a linear growth condition as well as for more general Lévy processes \(Z\).
In the second main result of the paper an approximation of the above stochastic differential equation by an equation driven by a Brownian motion when \(Z\) exhibits only small jumps is considered. Here the driving process is chosen such that the first and second moments coincide. A strong error estimate is derived under the same conditions on the coefficients as in the first result.
The paper is concluded with a numerical example.

MSC:

60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J75 Jump processes (MSC2010)
65C30 Numerical solutions to stochastic differential and integral equations
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