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)
StochasticDiffEq.SOSRI
— TypeSOSRI()
SOSRI: Stability-Optimized SRI Method (Nonstiff) - Recommended
The Stability-Optimized Stochastic Runge-Kutta method. This is the recommended method for general-purpose solving of diagonal/scalar Itô SDEs.
Method Properties
- Strong Order: 1.5 (for diagonal/scalar noise)
- Weak Order: 2.0
- Time stepping: Adaptive
- Noise types: Diagonal and scalar noise only
- SDE interpretation: Itô
- Stability: Optimized for high tolerances and robust to mild stiffness
When to Use
- Recommended as first choice for diagonal/scalar Itô SDEs
- When high accuracy is required (strong order 1.5)
- For problems with mild stiffness
- When using high tolerances (method is stable)
- For most general SDE applications
Algorithm Description
SOSRI is a stability-optimized version of the SRI methods with specially chosen coefficients to improve stability properties. It provides excellent performance for the most common class of SDE problems.
Restrictions
- Only works with diagonal or scalar noise
- For non-diagonal noise, use other methods like
RKMilCommute
orLambaEM
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
SOSRA - Stability-Optimized SRA (Optimal for Additive Noise)
StochasticDiffEq.SOSRA
— TypeSOSRA()
SOSRA: Stability-Optimized SRA Method (Nonstiff) - Optimal for Additive Noise
Stability-optimized adaptive Stochastic Runge-Kutta method for additive noise problems. This is the optimal choice for additive noise SDEs.
Method Properties
- Strong Order: 1.5 (for additive noise)
- Weak Order: 2.0
- Time stepping: Adaptive
- Noise types: Additive noise (diagonal, non-diagonal, and scalar)
- SDE interpretation: Both Itô and Stratonovich
- Stability: Optimized for high tolerances and robust to stiffness
When to Use
- Optimal choice for additive noise problems: du = f(u,p,t)dt + σ dW
- When the diffusion term is independent of the solution u
- For problems requiring high accuracy with additive noise
- When using high tolerances (method is stable)
- For both Itô and Stratonovich interpretations
Algorithm Description
SOSRA is a stability-optimized version of the SRA (Stochastic Runge-Kutta for Additive noise) methods. It exploits the special structure of additive noise to achieve better performance and stability.
Additive Noise Structure
Specialized for SDEs of the form:
du = f(u,p,t)dt + σ(t) dW
where the diffusion σ does not depend on the solution u.
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
Alternative SRI Methods
SRIW1 - SRI Weak Order 2
StochasticDiffEq.SRIW1
— TypeRößler A., Runge–Kutta Methods for the Strong Approximation of Solutions of Stochastic Differential Equations, SIAM J. Numer. Anal., 48 (3), pp. 922–952. DOI:10.1137/09076636X
SRIW1()
SRIW1: Stochastic Runge-Kutta W1 Method (Nonstiff)
Adaptive stochastic Runge-Kutta method with strong order 1.5 and weak order 2.0 for diagonal/scalar Itô SDEs.
Method Properties
- Strong Order: 1.5 (for diagonal/scalar noise)
- Weak Order: 2.0
- Time stepping: Adaptive
- Noise types: Diagonal and scalar noise only
- SDE interpretation: Itô
When to Use
- Standard choice for diagonal/scalar Itô SDEs
- When proven theoretical properties are important
- Alternative to SOSRI when stability optimization is not needed
- For problems requiring exactly weak order 2.0
Algorithm Features
- Embedded error estimation for adaptive stepping
- Well-established theoretical foundation
- Good balance of accuracy and efficiency
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
SRIW2 - SRI Weak Order 3
StochasticDiffEq.SRIW2
— TypeRößler A., Runge–Kutta Methods for the Strong Approximation of Solutions of Stochastic Differential Equations, SIAM J. Numer. Anal., 48 (3), pp. 922–952. DOI:10.1137/09076636X
SRIW2()
SRIW2: Stochastic Runge-Kutta W2 Method (Nonstiff)
Adaptive stochastic Runge-Kutta method with strong order 1.5 and weak order 3.0 for diagonal/scalar Itô SDEs.
Method Properties
- Strong Order: 1.5 (for diagonal/scalar noise)
- Weak Order: 3.0
- Time stepping: Adaptive
- Noise types: Diagonal and scalar noise only
- SDE interpretation: Itô
When to Use
- When weak order 3.0 convergence is required
- For Monte Carlo simulations needing high weak accuracy
- Problems where weak convergence is more important than strong
- When computational cost per step is acceptable for higher weak order
Algorithm Features
- Highest weak order in the SRI family
- More expensive per step than SRIW1
- Excellent for statistical calculations and expectations
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
SOSRI2 - Alternative Stability-Optimized SRI
StochasticDiffEq.SOSRI2
— TypeSOSRI2()
SOSRI2: Alternative Stability-Optimized SRI Method (Nonstiff)
Alternative stability-optimized adaptive strong order 1.5 method with different stability characteristics than SOSRI.
Method Properties
- Strong Order: 1.5 (for diagonal/scalar noise)
- Weak Order: 2.0
- Time stepping: Adaptive
- Noise types: Diagonal and scalar noise only
- SDE interpretation: Itô
- Stability: Optimized for high tolerances and robust to stiffness
When to Use
- Alternative to SOSRI with different stability properties
- When SOSRI performance is unsatisfactory
- For benchmarking stability-optimized methods
- Problems requiring different stability characteristics
Algorithm Features
- Different stability optimization than SOSRI
- May perform better on certain problem types
- Maintains high tolerance robustness
References
- Stability-optimized SRI methods
Alternative SRA Methods
SRA1 - Original SRA Method
StochasticDiffEq.SRA1
— TypeRößler A., Runge–Kutta Methods for the Strong Approximation of Solutions of Stochastic Differential Equations, SIAM J. Numer. Anal., 48 (3), pp. 922–952. DOI:10.1137/09076636X
SRA1()
SRA1: Stochastic Runge-Kutta A1 Method (Nonstiff)
Adaptive strong order 1.5 method for additive Itô and Stratonovich SDEs with weak order 2.
Method Properties
- Strong Order: 1.5 (for additive noise)
- Weak Order: 2.0
- Time stepping: Adaptive
- Noise types: Additive noise (diagonal, non-diagonal, and scalar)
- SDE interpretation: Both Itô and Stratonovich
When to Use
- Standard choice for additive noise problems
- When proven theoretical properties are important
- Alternative to SOSRA when stability optimization is not needed
- For both Itô and Stratonovich problems with additive noise
Additive Noise Structure
Specialized for SDEs of the form:
du = f(u,p,t)dt + σ(p,t) dW
where diffusion σ doesn't depend on solution u.
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
SRA2 - SRA Method Version 2
StochasticDiffEq.SRA2
— TypeRößler A., Runge–Kutta Methods for the Strong Approximation of Solutions of Stochastic Differential Equations, SIAM J. Numer. Anal., 48 (3), pp. 922–952. DOI:10.1137/09076636X
SRA2()
SRA2: Stochastic Runge-Kutta A2 Method (Nonstiff)
Alternative adaptive strong order 1.5 method for additive noise problems with different coefficients.
Method Properties
- Strong Order: 1.5 (for additive noise)
- Weak Order: 2.0
- Time stepping: Adaptive
- Noise types: Additive noise (diagonal, non-diagonal, and scalar)
- SDE interpretation: Both Itô and Stratonovich
When to Use
- Alternative to SRA1 with different stability/accuracy characteristics
- When SRA1 performance is unsatisfactory
- For benchmarking different SRA variants
- Research and comparison studies
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
SRA3 - SRA Method with Weak Order 3
StochasticDiffEq.SRA3
— TypeRößler A., Runge–Kutta Methods for the Strong Approximation of Solutions of Stochastic Differential Equations, SIAM J. Numer. Anal., 48 (3), pp. 922–952. DOI:10.1137/09076636X
SRA3()
SRA3: Stochastic Runge-Kutta A3 Method (Nonstiff)
Adaptive strong order 1.5 method for additive noise problems with weak order 3.
Method Properties
- Strong Order: 1.5 (for additive noise)
- Weak Order: 3.0
- Time stepping: Adaptive
- Noise types: Additive noise (non-diagonal and scalar)
- SDE interpretation: Both Itô and Stratonovich
When to Use
- When weak order 3.0 convergence is required for additive noise
- For Monte Carlo simulations needing highest weak accuracy
- Problems where weak convergence dominates computational cost
- When computational cost per step is acceptable for higher weak order
Restrictions
- Does not handle diagonal additive noise (use SRA1/SRA2 instead)
- Limited to non-diagonal and scalar additive noise structures
Algorithm Features
- Highest weak order in the SRA family
- More expensive per step than SRA1/SRA2
- Excellent for statistical calculations requiring high weak accuracy
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
SOSRA2 - Alternative Stability-Optimized SRA
StochasticDiffEq.SOSRA2
— TypeSOSRA2()
SOSRA2: Stability-Optimized SRA Method Version 2 (Nonstiff)
Alternative stability-optimized adaptive SRA method for additive noise problems.
Method Properties
- Strong Order: 1.5 (for additive noise)
- Weak Order: 2.0
- Time stepping: Adaptive
- Noise types: Additive noise (diagonal, non-diagonal, and scalar)
- SDE interpretation: Both Itô and Stratonovich
- Stability: Optimized for high tolerances and robust to stiffness
When to Use
- Alternative to SOSRA for additive noise problems
- Different stability characteristics may be preferred for specific problems
- When SOSRA performance is unsatisfactory
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
Configurable Methods
SRA - Configurable SRA with Custom Tableaux
StochasticDiffEq.SRA
— TypeRößler A., Runge–Kutta Methods for the Strong Approximation of Solutions of Stochastic Differential Equations, SIAM J. Numer. Anal., 48 (3), pp. 922–952. DOI:10.1137/09076636X
SRA(;tableau=constructSRA1())
SRA: Configurable Stochastic Runge-Kutta for Additive Noise (Nonstiff)
Configurable adaptive strong order 1.5 method for additive noise problems with customizable tableaux.
Method Properties
- Strong Order: 1.5 (for additive noise)
- Weak Order: Depends on tableau (typically 2.0)
- Time stepping: Adaptive
- Noise types: Additive noise (diagonal, non-diagonal, and scalar)
- SDE interpretation: Both Itô and Stratonovich
Parameters
tableau
: Tableau specification (default:constructSRA1()
)
When to Use
- When custom tableaux are needed for additive noise problems
- For research and experimentation with SRA methods
- When default methods don't provide desired characteristics
- For benchmarking different SRA variants
Available Tableaux
constructSRA1()
: Default SRA1 tableau- Custom tableaux can be constructed for specialized applications
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
SRI - Configurable SRI with Custom Tableaux
StochasticDiffEq.SRI
— TypeRößler A., Runge–Kutta Methods for the Strong Approximation of Solutions of Stochastic Differential Equations, SIAM J. Numer. Anal., 48 (3), pp. 922–952. DOI:10.1137/09076636X
SRI(;tableau=constructSRIW1(), error_terms=4)
SRI: Configurable Stochastic Runge-Kutta for Itô SDEs (Nonstiff)
Configurable adaptive strong order 1.5 method for diagonal/scalar Itô SDEs with customizable tableaux.
Method Properties
- Strong Order: 1.5 (for diagonal/scalar noise)
- Weak Order: Depends on tableau (typically 2.0)
- Time stepping: Adaptive
- Noise types: Diagonal and scalar noise only
- SDE interpretation: Itô
Parameters
tableau
: Tableau specification (default:constructSRIW1()
)error_terms::Int = 4
: Number of error terms for adaptive stepping
When to Use
- When custom tableaux are needed for diagonal/scalar problems
- For research and experimentation with SRI methods
- When default methods don't provide desired characteristics
- For benchmarking different SRI variants
Available Tableaux
constructSRIW1()
: Default SRIW1 tableau- Custom tableaux can be constructed for specialized applications
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
Method Selection Guide
For Diagonal/Scalar Noise:
- First choice: SOSRI - Best overall performance and stability
- Alternative: SRIW1 - Standard SRI method
- High weak order: SRIW2 - When weak order 3 is needed
For Additive Noise:
- First choice: SOSRA - Optimal for additive noise structure
- Alternative: SRA1 - Standard SRA method
- 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