The approximate Euler method for Lévy driven stochastic differential equations

By Jean Jacod, Thomas G. Kurtz, S. M'el'eard, Philip Protter. 2005

98 citations recorded by the source corpus.

Abstract This paper is concerned with the numerical approximation of the expected value E ( g ( X t ) ) , where g is a suitable test function and X is the solution of a stochastic differential equation driven by a Levy process Y. More precisely we consider an Euler scheme or an “approximate” Euler scheme with stepsize 1 / n , giving rise to a simulable variable X t n , and we study the error δ n ( g ) = E ( g ( X t n ) ) − E ( g ( X t ) ) . For a genuine Euler scheme we typically get that δ n ( g ) is of order 1 / n , and we even have an expansion of this error in successive powers of 1 / n , and the assumptions are some integrability condition on the driving process and appropriate smoothness of the coefficient of the equation and of the test function g. For an approximate Euler scheme, that is we replace the non-simulable increments of X by a simulable variable close enough to the desired increment, the order of magnitude of δ n ( g ) is the supremum of 1 / N and a kind of “distance” between the increments of Y and the actually simulated variable. In this situation, a second order expansion is also available.

Subfields: Finance, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/859bc8364d09ecb50bfdfd9d58b214c325ed75c7

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Citations in the corpus

A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations (2011, methodology): “We refer in particular to [PT97], [Rub03], and [JKMP05] for an analysis of the Euler scheme for Lévy-driven SDEs.”
Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing (2005, background): “For stochastic differential equations driven by a Lévy process, see [8].”
Testing for Common Arrivals of Jumps for Discretely Observed Multidimensional Processes (2009, unknown): “1 of [10], with Ht = 1 and at = t), plus the obvious properties ′(ε) ≤ K and ∫Aε γ (z)λ(dz) ≤ K ′(ε), give us E(Z(ε) p/2 t ) ≤ Kt ′(ε) when t ∈ [0,1].”
Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise (2011, unknown): “, [20, 35, 39, 42] and the references therein.”
Simulation and approximation of Lévy-driven stochastic differential equations (2009, unknown): “Jacod-Kurtz-Méléard-Protter [8] have computed systematically the weak error for the approximate Euler scheme.”
Jump-adapted discretization schemes for Lévy-driven SDEs (2010, methodology): “The traditional method to simulate X is to use the Euler scheme which is a uniformly-spaced discretization scheme for (1) [19, 12].”
On tamed milstein schemes of SDEs driven by Lévy noise (2014, background): “The proof is an immediate consequence of Lemmas [11, 12].”
Parameter estimation for Ornstein-Uhlenbeck processes driven by α-stable Lévy motions (2007, methodology): “The Euler scheme for stochastic differential equations driven by Lévy processes has been well studied (see [5], [6], [7], [13] and the references therein).”