Strong approximations of stochastic differential equations with jumps

By Nicola Bruti-Liberati, Eckhard Platen. 2007

88 citations recorded by the source corpus.

This paper is a survey of strong discrete time approximations of jump-diffusion processes described by stochastic differential equations (SDEs). It also presents new results on strong discrete time approximations for the specific case of pure jump SDEs. Strong approximations based on jump-adapted time discretizations, which produce no discretization error in the case of pure jump processes, are analyzed. The computational complexity of these approximations is proportional to the jump intensity. By exploiting a stochastic expansion for pure jump processes, higher order discrete time approximations, whose computational complexity is not dependent on the jump intensity, are proposed. For the specific case of pure jump SDEs, the strong order of convergence of strong Taylor schemes is established under weaker conditions than those currently known in the literature.

Subfields: Computational Theory and Mathematics, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/de451a4f69da944a1bd1931898f806caf5af1b8b

How researchers see this paper

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In Computational Theory and Mathematics

1. V. Todorov
2. Peter Tankov
3. Sotirios Sabanis

In Finance

1. Chaman Kumar
2. Arturo Kohatsu-Higa
3. V. Todorov
4. Peter Tankov
5. Sotirios Sabanis
6. Tim Bollerslev
7. Xuerong Mao
8. Andrew J. Patton

In Statistics and Probability

1. Arturo Kohatsu-Higa
2. V. Todorov
3. Peter Tankov
4. Sotirios Sabanis
5. Peter J. Brockwell
6. Ekaterina Voltchkova
7. Rama Cont
8. A. Kulik

Citations in the corpus

• Estimation of continuous-time stochastic volatility models with jumps using high-frequency data (2009, methodology): “This is the Non-Gaussian OU model of Barndorff-Nielsen and Shep(4)Corradi and Distaso (2006) analyze only the case when IV is used in the estimation.”
• On tamed milstein schemes of SDEs driven by Lévy noise (2014, background): “The proof is an immediate consequence of Lemmas [11, 12].”
• Jump-adapted discretization schemes for Lévy-driven SDEs (2010, background): “A natural idea due to Rubenthaler [21] (in the context of finite-intensity jump processes, this idea appears also in [3, 16]), is to replace the process Z with a suitable compound Poisson approximation and place the discretization points at the jump times of the compound Poisson process.”