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.

SOSRI - Stability-Optimized SRI (Recommended)

StochasticDiffEq.SOSRI — Type
SOSRI()

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 or LambaEM

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 — Type
SOSRA()

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 — Type

Rößler 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 — Type

Rößler 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 — Type
SOSRI2()

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 — Type

Rößler 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 — Type

Rößler 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 — Type

Rößler 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 — Type
SOSRA2()

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 — Type

Rößler 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 — Type

Rößler 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:

  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