[Paper Review] On Tamed Euler Approximations of SDEs Driven by Lévy Noise with Applications to Delay Equations

Konstantinos Dareiotis, Chaman Kumar+1 more|White Rose Research Online (University of Leeds, The University of Sheffield, University of York)|Mar 3, 2014

Stochastic processes and financial applications10 references4 citations

TL;DR

This paper introduces tamed Euler schemes for stochastic differential equations (SDEs) driven by Lévy noise with super-linearly growing drift coefficients, extending taming techniques to handle jumps. It establishes strong $\mathcal{L}^q$ convergence under local Lipschitz conditions and provides rate of convergence results under global and polynomial Lipschitz assumptions, further extending the framework to stochastic delay equations (SDDEs) with non-Markovian, delayed dynamics.

ABSTRACT

We extend the taming techniques for explicit Euler approximations of stochastic differential equations (SDEs) driven by Lévy noise with super-linearly growing drift coefficients. Strong convergence results are presented for the case of locally Lipschitz coefficients. Moreover, rate of convergence results are obtained in agreement with classical literature when the local Lipschitz continuity assumptions are replaced by global and, in addition, the drift coefficients satisfy polynomial Lipschitz continuity. Finally, we further extend these techniques to the case of delay equations.

Motivation & Objective

• Address the lack of strong convergence in explicit Euler schemes for SDEs with Lévy noise when drift coefficients grow super-linearly.
• Extend taming techniques—previously used for diffusion SDEs—to SDEs driven by Lévy noise, enabling numerical approximation of such equations.
• Establish strong convergence in $\mathcal{L}^q$ norm for SDEs with locally Lipschitz coefficients and derive convergence rates under global and polynomial Lipschitz conditions.
• Extend the tamed Euler framework to stochastic delay differential equations (SDDEs) driven by Lévy noise with super-linear drift in both non-delayed and delayed variables.
• Provide existence and uniqueness results for SDDEs under relaxed local Lipschitz conditions on non-delayed variables only, improving upon existing literature.

Proposed method

• Develop a tamed Euler scheme by modifying the drift coefficient via a taming function that prevents explosive growth, ensuring stability in the numerical scheme.
• Apply the taming technique from [21] to SDEs driven by Lévy noise, using a truncation mechanism based on the inverse of the norm of the drift coefficient.
• Use a piecewise-constant approximation of the solution process over time intervals, with tamed drift terms that depend on the previous step's state.
• Employ a recursive argument over time intervals to prove convergence, relying on moment bounds and Gronwall-type inequalities.
• Utilize the approach of [7] to link delay equations to random coefficient SDEs, enabling the extension of tamed schemes to SDDEs.
• Apply Hölder’s inequality, moment estimates, and inductive arguments to control the error between the true solution and the tamed scheme.

Experimental results

Research questions

• RQ1 Can tamed Euler schemes be extended to SDEs driven by Lévy noise with super-linearly growing drift coefficients, where standard Euler schemes fail to converge?
• RQ2 What conditions on the drift, diffusion, and jump coefficients ensure strong $\mathcal{L}^q$ convergence of the tamed Euler scheme for SDEs with Lévy noise?
• RQ3 What is the rate of convergence of the tamed Euler scheme when global and polynomial Lipschitz conditions are imposed on the coefficients?
• RQ4 Can the taming framework be adapted to stochastic delay differential equations (SDDEs) with jumps and super-linear drift in both delayed and non-delayed components?
• RQ5 Under what relaxed regularity conditions—specifically, local Lipschitz continuity only in non-delayed variables—can existence and uniqueness of solutions to SDDEs with Lévy noise be established?

Key findings

• The tamed Euler scheme achieves strong $\mathcal{L}^q$ convergence for SDEs with Lévy noise under one-sided local Lipschitz and local Lipschitz conditions on drift, diffusion, and jump coefficients.
• For globally Lipschitz and polynomially Lipschitz drift coefficients, the scheme attains a convergence rate of order $O(n^{-1/2})$ in $\mathcal{L}^q$ norm, consistent with classical results.
• The method is extended to SDDEs with super-linear drift in both non-delayed and delayed variables, with coefficients satisfying one-sided Lipschitz and polynomial Lipschitz conditions.
• The convergence rate for SDDEs is $O(n^{-1/2})$ under global and polynomial Lipschitz assumptions on drift, diffusion, and jump coefficients.
• The paper proves existence and uniqueness of solutions to SDDEs under relaxed conditions: local Lipschitz continuity only in non-delayed variables, with continuity in delayed variables.
• Numerical experiments confirm the theoretical convergence rates: for SDE (78), the $\mathcal{L}^2$ error decreases approximately as $O(n^{-1/2})$ with step size $h = 2^{-k}$, and similar behavior is observed for SDDE (79).