论文信息 - Approximations of small jumps of Lévy processes with a view towards simulation

Approximations of small jumps of Lévy processes with a view towards simulation

Let X = (X(t):t ≥ 0) be a Lévy process and X ∊ the compensated sum of jumps not exceeding ∊ in absolute value, σ2(∊) = var(X ∊(1)). In simulation, X - X ∊ is easily generated as the sum of a Brownian term and a compound Poisson one, and we investigate here when X ∊/σ(∊) can be approximated by another Brownian term. A necessary and sufficient condition in terms of σ(∊) is given, and it is shown that when the condition fails, the behaviour of X ∊/σ(∊) can be quite intricate. This condition is also related to the decay of terms in series expansions. We further discuss error rates in terms of Berry-Esseen bounds and Edgeworth approximations.

S. Asmussen | J. Rosínski

[1] Michael B. Marcus,et al. $L^1$-Norm of Infinitely Divisible Random Vectors and Certain Stochastic Integrals , 2001 .

[2] J. Rosínski. Series Representations of Lévy Processes from the Perspective of Point Processes , 2001 .

[3] V. Statulevičius,et al. Limit Theorems of Probability Theory , 2000 .

[4] Ole E. Barndorff-Nielsen,et al. Processes of normal inverse Gaussian type , 1997, Finance Stochastics.

[5] Tina Hviid Rydberg. The normal inverse gaussian lévy process: simulation and approximation , 1997 .

[6] L. Heinrich,et al. Normal and poisson approximation of infinitely divisible distribution functions , 1991 .

[7] Ib M. Skovgaard,et al. On Multivariate Edgeworth Expansions , 1986 .

[8] F. Götze. Asymptotic expansions in functional limit theorems , 1985 .

[9] L. Bondesson. On simulation from infinitely divisible distributions , 1982, Advances in Applied Probability.

[10] C. Mallows,et al. A Method for Simulating Stable Random Variables , 1976 .

[11] R. Rao,et al. Normal Approximation and Asymptotic Expansions , 1976 .

[12] J. Lamperti. ON CONVERGENCE OF STOCHASTIC PROCESSES , 1962 .