Adaptive Euler-Maruyama Scheme for Stochastic Delay Differential Equations with Non-Global Lipschitz Drift

Core Concepts

This paper proposes an adaptive Euler-Maruyama numerical method for solving stochastic delay differential equations (SDDEs) with non-global Lipschitz drift terms and non-constant delays. The method adapts the step size based on the growth of the drift term, overcoming the challenge of numerical nodes not falling within the nodes after subtracting the delay.

Abstract

The paper presents an adaptive Euler-Maruyama scheme for solving stochastic delay differential equations (SDDEs) with non-global Lipschitz drift terms and non-constant delays. The key highlights are:

The proposed method adapts the step size based on the growth of the drift term, which relaxes the coefficient requirements compared to the fixed-step Euler-Maruyama method.
The paper addresses the challenge of numerical nodes not falling within the nodes after subtracting the delay by substituting the delay term with the numerically obtained solution closest to the left endpoint.
The authors prove the convergence of the numerical method for a class of non-global Lipschitz continuous SDDEs under the assumption that the step size function satisfies certain conditions.
The paper establishes the stability and strong convergence of the proposed adaptive scheme, including the strong convergence order.
An estimate of the expected number of time steps per path is provided, showing that it increases linearly with the time interval, similar to the case of uniform time steps.

The proposed adaptive method expands the applicability of numerical schemes for SDDEs by relaxing the coefficient requirements and handling non-constant delays effectively.

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Key Insights Distilled From

An adaptive Euler-Maruyama scheme for SDDEs

by Dongyang Liu... at arxiv.org 04-17-2024

https://arxiv.org/pdf/2404.10244.pdf

Deeper Inquiries

How can the proposed adaptive Euler-Maruyama scheme be extended to handle more general forms of SDDEs, such as those with state-dependent delays or non-Lipschitz coefficients

The proposed adaptive Euler-Maruyama scheme can be extended to handle more general forms of SDDEs by incorporating additional features and modifications to accommodate the complexities of state-dependent delays or non-Lipschitz coefficients.
For SDDEs with state-dependent delays, the adaptive scheme can be adjusted to dynamically update the step size based on the current state and delay values. This adaptation can involve incorporating feedback mechanisms that adjust the step size based on the evolution of the system, ensuring accurate and stable numerical solutions. By considering the state-dependent nature of delays, the scheme can effectively capture the dynamic behavior of the system and provide more accurate approximations.
In the case of non-Lipschitz coefficients, the adaptive scheme can be enhanced by employing regularization techniques or alternative numerical integration methods that are suitable for handling non-smooth functions. By incorporating regularization terms or utilizing specialized integration algorithms designed for non-Lipschitz functions, the adaptive scheme can effectively handle the challenges posed by such coefficients and ensure the stability and convergence of the numerical solutions.
Overall, by incorporating tailored adjustments and enhancements to the adaptive Euler-Maruyama scheme, it can be extended to address a broader range of SDDEs with varying complexities, including those with state-dependent delays or non-Lipschitz coefficients.

What are the potential challenges and limitations of the adaptive step size control approach when dealing with high-dimensional SDDEs or SDDEs with more complex delay structures

The adaptive step size control approach, while effective in handling stochastic delay differential equations (SDDEs), may face challenges and limitations when dealing with high-dimensional SDDEs or those with more complex delay structures. Some potential challenges and limitations include:

Computational Complexity: High-dimensional SDDEs require significant computational resources to adaptively adjust the step size at each iteration. The increased dimensionality can lead to longer computation times and higher memory requirements, impacting the efficiency of the numerical method.

Convergence Issues: In high-dimensional systems, the adaptive step size control may struggle to maintain stability and convergence, especially when dealing with intricate delay structures. The complexity of the system dynamics can introduce numerical instabilities that affect the accuracy of the solutions.

Optimal Step Size Selection: Determining the optimal step size in high-dimensional SDDEs with complex delay structures can be challenging. The adaptive control mechanism may need to strike a balance between accuracy and computational efficiency, requiring careful tuning of parameters.

Adaptation to Nonlinearities: Complex delay structures and nonlinearity in high-dimensional SDDEs can pose challenges for the adaptive step size control approach. Adapting the step size to nonlinear dynamics and intricate delay dependencies may require sophisticated algorithms and strategies.

Addressing these challenges and limitations may involve developing specialized techniques for adaptive step size control in high-dimensional SDDEs, incorporating advanced numerical methods, and optimizing the algorithm parameters to ensure robust and efficient solutions.

Can the insights from this work on adaptive numerical methods for SDDEs be leveraged to develop efficient algorithms for solving other types of stochastic differential equations with delay, such as those arising in finance, biology, or engineering applications

The insights from this work on adaptive numerical methods for SDDEs can indeed be leveraged to develop efficient algorithms for solving other types of stochastic differential equations with delay in various fields such as finance, biology, or engineering applications. Some ways in which these insights can be applied include:

Finance: In finance, where stochastic processes with delays are prevalent in modeling asset prices or interest rate dynamics, adaptive numerical methods can enhance the accuracy and efficiency of pricing models, risk management tools, and derivative pricing algorithms. By incorporating adaptive step size control, financial models can better capture the dynamics of market fluctuations and improve decision-making processes.

Biology: Biological systems often exhibit complex dynamics with time delays, making SDDEs a valuable tool for modeling biological processes such as gene regulation, population dynamics, or disease spread. Adaptive numerical methods can aid in simulating and analyzing these systems, providing insights into the impact of delays on biological phenomena and facilitating the development of predictive models for various biological scenarios.

Engineering: In engineering applications, where systems with delays are common in control systems, signal processing, or communication networks, adaptive numerical methods can optimize system performance, reduce delays, and enhance stability. By applying adaptive step size control techniques, engineers can design more robust and efficient systems that account for delays and uncertainties in real-time applications.

By leveraging the principles and methodologies of adaptive numerical methods for SDDEs, researchers and practitioners in finance, biology, and engineering can develop tailored algorithms to address specific challenges in their respective fields, leading to improved modeling accuracy, predictive capabilities, and system performance.