SRA/SRI Methods - Stochastic Runge-Kutta

The SRA (Stochastic Runge-Kutta for Additive noise) and SRI (Stochastic Runge-Kutta for Itô) methods provide high-order adaptive solvers for different noise structures. These are among the most effective methods for their respective problem classes.

Recommended Methods

SOSRI - Stability-Optimized SRI (Recommended)

SOSRA - Stability-Optimized SRA (Optimal for Additive Noise)

Alternative SRI Methods

SRIW1 - SRI Weak Order 2

SRIW2 - SRI Weak Order 3

SOSRI2 - Alternative Stability-Optimized SRI

Alternative SRA Methods

SRA1 - Original SRA Method

SRA2 - SRA Method Version 2

SRA3 - SRA Method with Weak Order 3

SOSRA2 - Alternative Stability-Optimized SRA

Configurable Methods

SRA - Configurable SRA with Custom Tableaux

SRI - Configurable SRI with Custom Tableaux

Method Selection Guide

For Diagonal/Scalar Noise:

1. First choice: SOSRI - Best overall performance and stability
2. Alternative: SRIW1 - Standard SRI method
3. High weak order: SRIW2 - When weak order 3 is needed

For Additive Noise:

1. First choice: SOSRA - Optimal for additive noise structure
2. Alternative: SRA1 - Standard SRA method
3. High weak order: SRA3 - When weak order 3 is needed

Performance Characteristics:

• SOSRI/SOSRA: Stability-optimized, robust to high tolerances
• SRIWx/SRAx: Standard methods with proven theoretical properties
• SRA/SRI: Allow custom tableaux for specialized applications

Theoretical Foundation

The SRA and SRI methods are based on stochastic Runge-Kutta theory:

SRA Methods exploit the additive noise structure:

du = f(u,t)dt + σ(t)dW

Where the diffusion σ doesn't depend on the solution u.

SRI Methods handle the general diagonal case:

du = f(u,t)dt + g(u,t)dW

Where each component has independent noise.

Both method families achieve:

• Strong order 1.5 convergence
• Weak order 2.0 or higher
• Adaptive time stepping with embedded error estimation
• A-stable or L-stable properties (for optimized versions)

References

• Rößler A., "Runge–Kutta Methods for the Strong Approximation of Solutions of Stochastic Differential Equations", SIAM J. Numer. Anal., 48 (3), pp. 922–952