Strong Convergence Analysis of the Exponential Euler Scheme for SDEs with Superlinear Growth Coefficients and Piecewise Locally Lipschitz Drift

Core Concepts

The exponential Euler (Exp-EM) scheme is shown to have a strong convergence rate of 1/2 for SDEs with superlinear growth coefficients and piecewise locally Lipschitz drift, under certain parameter conditions.

Abstract

The paper analyzes the strong convergence of the exponential Euler (Exp-EM) scheme for the numerical approximation of the solution to a one-dimensional stochastic differential equation (SDE) with piecewise locally Lipschitz drift and continuous diffusion coefficients with polynomial growth.
Key highlights:

The Exp-EM scheme is introduced, which preserves the positivity of the solution and maintains identical finite moments as the exact solution under certain conditions.
Sufficient conditions are provided for the strong wellposedness of the SDE solution, including the control of negative, positive, and exponential moments.
The local error of the Exp-EM scheme is analyzed, establishing a 1/2 rate of convergence for the Lp-norm of the error process, where the rate is penalized by a factor ε when the drift is discontinuous.
A change of time technique is employed to circumvent the stochastic Gronwall estimation step and obtain the strong convergence rate.
The asymptotic behavior and theoretical stability of the Exp-EM scheme are also explored, and numerical experiments are presented to validate the theoretical findings.

Stats

The diffusion coefficient σ satisfies |σ(x)|^2 ≤ Σ^2 x^(2α), for x ∈ R+, where Σ > 0 and α > 1.
The drift b satisfies b(x) ≤ b(0) + B1 x - B2 x^β, for x ∈ R+, where β ≥ 2α - 1 > 1, and b(0), B1, B2 are non-negative constants.

Quotes

"We show the usual 1/2 rate of convergence for the exponential-Euler scheme when the drift is continuous. When the drift is discontinuous, the convergence rate is penalised by a factor ε decreasing with the time-step."
"We examine the case of the diffusion coefficient vanishing at zero, which adds a positivity preservation condition and a convergence analysis that exploits the negative moments and exponential moments of the scheme with the help of change of time technique introduced in Berkaoui et al. (2008)."

Key Insights Distilled From

Strong convergence of the exponential Euler scheme for SDEs with superlinear growth coefficients and one-sided Lipschitz drift

by Mire... at arxiv.org 05-03-2024

https://arxiv.org/pdf/2405.00806.pdf

Deeper Inquiries

How can the Exp-EM scheme be extended to handle higher-dimensional SDEs with more general diffusion coefficient classes

To extend the Exp-EM scheme to handle higher-dimensional SDEs with more general diffusion coefficient classes, we can follow a similar approach as in the one-dimensional case. In the multidimensional setting, the diffusion coefficient can be a d×r matrix, consistent with the r-dimensional Brownian motion. The Exp-EM scheme for multidimensional SDEs can be expressed as:
[dX_{t}^{i} = X_{t}^{i} \left( \frac{b(X_{t}) - b(0)}{X_{t}} dt + \frac{\sigma(X_{t}) - \sigma(0)}{X_{t}} dW_{t} \right) + b(0) dt + \sigma(0) dW_{t}]
This formulation allows for the handling of SDEs in higher dimensions, where each component of the process follows a similar exponential Euler scheme. The key is to ensure that the scheme preserves positivity and moments in each dimension, similar to the one-dimensional case.

What are the potential challenges and considerations in the convergence analysis for the multidimensional case, especially when the drift has a more complex discontinuity structure

In the convergence analysis for the multidimensional case, especially when the drift has a more complex discontinuity structure, there are several potential challenges and considerations:

Discontinuity Handling: The presence of discontinuities in the drift function in higher dimensions can complicate the convergence analysis. Special attention needs to be paid to the topology of the discontinuity sets and how they affect the approximation scheme.

Occupation Time Estimation: Techniques like occupation time formulas and Bernstein's inequality, used in the one-dimensional case, may need to be extended to the multidimensional setting to handle discontinuities effectively.

Control of Exponential Moments: Controlling exponential moments for the multidimensional Exp-EM scheme becomes more challenging with complex drift structures. Ensuring the necessary conditions for convergence, especially in the presence of discontinuities, requires careful analysis.

Adaptation to Higher Dimensions: The extension to higher dimensions introduces additional complexity in terms of computational efficiency and stability. Techniques for handling multidimensional processes need to be carefully adapted to ensure accurate convergence analysis.

Can the techniques developed in this work be adapted to design adaptive time-stepping strategies for the Exp-EM scheme to handle increasingly explosive cases of SDEs

The techniques developed in this work can be adapted to design adaptive time-stepping strategies for the Exp-EM scheme to handle increasingly explosive cases of SDEs. Here are some ways this can be achieved:

Threshold-based Adaptation: Similar to the threshold used in the convergence analysis, adaptive time-stepping strategies can use thresholds to dynamically adjust the time step based on the behavior of the process. When the process approaches certain critical values, the time step can be refined to ensure accuracy.

Local Error Estimation: By monitoring the local error of the scheme, adaptive strategies can detect when the approximation deviates significantly from the exact solution. This information can be used to dynamically adjust the time step to maintain accuracy.

Dynamic Parameter Control: Adaptive strategies can also incorporate dynamic control of parameters such as the drift and diffusion coefficients based on the behavior of the process. This adaptive parameter tuning can help improve the stability and convergence of the scheme in explosive cases.

Multilevel Methods: Leveraging multilevel methods in combination with adaptive time-stepping can further enhance the efficiency and accuracy of the Exp-EM scheme in handling complex SDEs with superlinear growth coefficients and discontinuities.