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Modify the Improved Euler scheme to integrate stochastic differential equations

A. J. Roberts

TL;DR

The paper introduces a Runge-Kutta–type scheme for Itô SDEs that closely mirrors the deterministic Improved Euler method while incorporating stochastic increments. It proves strong convergence of order one (O(h)) in general, with faster O(h^2) behavior in certain linear additive-noise cases, and it demonstrates the method numerically on several test SDEs. The analysis connects the proposed scheme to Milstein-type expansions and shows how it can readily adapt to Stratonovich SDEs by setting a sign parameter appropriately. The work offers a practical, entry-level tool for teaching stochastic dynamics and for simulations with small noise, while outlining limitations and directions for extending to multiple noise sources.

Abstract

A practical and new Runge--Kutta numerical scheme for stochastic differential equations is explored. Numerical examples demonstrate the strong convergence of the method. The first order strong convergence is then proved using Ito integrals for both Ito and Stratonovich interpretations. As a straightforward modification of the deterministic Improved Euler/Heun method, the method is a good entry level scheme for stochastic differential equations, especially in conjunction with Higham's introduction [SIAM Review, 43:525--546, 2001].

Modify the Improved Euler scheme to integrate stochastic differential equations

TL;DR

The paper introduces a Runge-Kutta–type scheme for Itô SDEs that closely mirrors the deterministic Improved Euler method while incorporating stochastic increments. It proves strong convergence of order one (O(h)) in general, with faster O(h^2) behavior in certain linear additive-noise cases, and it demonstrates the method numerically on several test SDEs. The analysis connects the proposed scheme to Milstein-type expansions and shows how it can readily adapt to Stratonovich SDEs by setting a sign parameter appropriately. The work offers a practical, entry-level tool for teaching stochastic dynamics and for simulations with small noise, while outlining limitations and directions for extending to multiple noise sources.

Abstract

A practical and new Runge--Kutta numerical scheme for stochastic differential equations is explored. Numerical examples demonstrate the strong convergence of the method. The first order strong convergence is then proved using Ito integrals for both Ito and Stratonovich interpretations. As a straightforward modification of the deterministic Improved Euler/Heun method, the method is a good entry level scheme for stochastic differential equations, especially in conjunction with Higham's introduction [SIAM Review, 43:525--546, 2001].

Paper Structure

This paper contains 10 sections, 29 equations, 4 figures.

Table of Contents

  1. Introduce a modified integration
  2. Examples demonstrate $\mathcal{O} (h)$ error is typical
  3. Autonomous example
  4. Non-autonomous example
  5. Example with second order error
  6. Prove $\mathcal{O} (h)$ global error in general
  7. Error for Itô integrals
  8. Error for linear SDEs with additive noise
  9. Global error for general SDEs
  10. Conclusion

Figures (4)

  • Figure 1: As the time step is successively halved, $n=16,32,64,128,256$ time steps over $0\leq t\leq1$ , the numerical solutions of the sde\ref{['eq:aeg']} via the method \ref{['eq:ieuabj']} appear to converge.
  • Figure 2: Average over $700$ realisations at each of 13 different step sizes for the sde\ref{['eq:aeg']}: at $t=1$ , the rms error in the predicted $X(1,\omega)$ decreases linearly in time step $h$.
  • Figure 3: Average over $700$ realisations at each of 13 different step sizes for the sde\ref{['eq:naeg']}: at $t=1$ , the rms error in the predicted $X(1,\omega)$ decreases linearly in time step $h$.
  • Figure 4: Averaging over $700$ realisations at each of 13 different step sizes for the linear sde\ref{['eq:qeg']}: at $t=1$ , the rms error in the predicted $X(1,\omega)$ decreases quadratically, like $h^{2}$.

Theorems & Definitions (4)

  • proof
  • proof
  • proof
  • proof