Commutative Noise Methods

When multiple noise sources satisfy commutativity conditions, specialized methods can achieve higher accuracy and efficiency compared to general methods. These methods avoid expensive Lévy area computations while maintaining high order.

RKMilCommute - Runge-Kutta Milstein for Commutative Noise

StochasticDiffEq.RKMilCommute — Type

Kloeden, P.E., Platen, E., Numerical Solution of Stochastic Differential Equations. Springer. Berlin Heidelberg (2011)

RKMilCommute(;interpretation=AlgorithmInterpretation.Ito, ii_approx=IICommutative())

RKMilCommute: Runge-Kutta Milstein for Commutative Noise (Nonstiff) - Recommended for Commutative Noise

Explicit Runge-Kutta discretization of the strong order 1.0 Milstein method specialized for commutative noise problems.

Method Properties

  • Strong Order: 1.0
  • Weak Order: Depends on tableau
  • Time stepping: Adaptive (1.5/2.0 error estimate)
  • Noise types: Commutative noise (multiple noise sources that commute)
  • SDE interpretation: Configurable (Itô or Stratonovich)

Parameters

  • interpretation: Choose AlgorithmInterpretation.Ito (default) or AlgorithmInterpretation.Stratonovich
  • ii_approx: Iterated integral approximation method (default: IICommutative())

When to Use

  • Recommended for commutative noise problems
  • When you have multiple noise sources that satisfy commutativity conditions
  • For multi-dimensional SDEs with commuting noise terms
  • When higher order accuracy than Euler-Maruyama is needed

Commutative Noise

Applicable when the noise terms satisfy:

[g_i, g_j] = g_i(∂g_j/∂x) - g_j(∂g_i/∂x) = 0

for all noise terms gi, gj.

Algorithm Description

Extends the Milstein method to handle multiple commutative noise sources efficiently without requiring the full Lévy area computation.

References

  • Kloeden, P.E., Platen, E., "Numerical Solution of Stochastic Differential Equations", Springer (1992)

RKMilGeneral - General Milstein for Non-commutative Noise

StochasticDiffEq.RKMilGeneral — Type

Kloeden, P.E., Platen, E., Numerical Solution of Stochastic Differential Equations. Springer. Berlin Heidelberg (2011)

RKMilGeneral(;interpretation=AlgorithmInterpretation.Ito, ii_approx=IILevyArea(), c=1, p=nothing, dt=nothing)

RKMilGeneral: General Runge-Kutta Milstein Method (Nonstiff)

Explicit Runge-Kutta discretization of the Milstein method for general non-commutative noise problems using Lévy area approximation.

Method Properties

  • Strong Order: 1.0 (for general noise)
  • Weak Order: Depends on tableau and Lévy area approximation
  • Time stepping: Adaptive
  • Noise types: All forms including non-commutative noise
  • SDE interpretation: Configurable (Itô or Stratonovich)

Parameters

  • interpretation: Choose AlgorithmInterpretation.Ito (default) or AlgorithmInterpretation.Stratonovich
  • ii_approx: Iterated integral approximation method (default: IILevyArea())
  • c::Int = 1: Truncation parameter for Lévy area
  • p: Truncation level (computed automatically if nothing)
  • dt: Used for automatic truncation level computation

When to Use

  • For general non-commutative noise problems
  • When RKMilCommute is not applicable (noise doesn't commute)
  • When higher accuracy than Euler methods is needed
  • For complex multi-dimensional noise structures

Lévy Area Approximation

Uses LevyArea.jl for efficient computation of iterated integrals:

  • Automatically selects optimal algorithms
  • Handles truncation for practical computation
  • Supports various approximation strategies

Computational Cost

  • More expensive than commutative methods
  • Lévy area computation scales with noise dimension
  • Adaptive truncation balances accuracy and efficiency

References

  • Kastner, F. and Rößler, A., "LevyArea.jl: A Julia package for Lévy area computation", arXiv:2201.08424
  • LevyArea.jl: https://github.com/stochastics-uni-luebeck/LevyArea.jl

Three-Stage Milstein Methods

WangLi3SMil Family - Fixed Step Milstein Methods

StochasticDiffEq.WangLi3SMil_A — Type
WangLi3SMil_A()

WangLi3SMil_A: 3-Stage Milstein Method A (Nonstiff)

Fixed step-size explicit 3-stage Milstein method with strong and weak order 1.0 for Itô SDEs.

Method Properties

  • Strong Order: 1.0
  • Weak Order: 1.0
  • Time stepping: Fixed step size
  • Noise types: Depends on tableau (typically diagonal/scalar)
  • SDE interpretation: Itô

When to Use

  • When fixed step size is preferred
  • For Itô SDEs requiring order 1.0 accuracy
  • Part of WangLi family - compare performance with other variants
  • When computational cost per step is less important than simplicity

References

  • Wang and Li, "Three-stage stochastic Runge-Kutta methods for stochastic differential equations"
StochasticDiffEq.WangLi3SMil_B — Type
WangLi3SMil_B()

WangLi3SMil_B: 3-Stage Milstein Method B (Nonstiff)

Alternative 3-stage Milstein method with different stability and accuracy characteristics.

Method Properties

  • Strong Order: 1.0
  • Weak Order: 1.0
  • Time stepping: Fixed step size
  • Noise types: Depends on tableau (typically diagonal/scalar)
  • SDE interpretation: Itô

References

  • Wang and Li, "Three-stage stochastic Runge-Kutta methods for stochastic differential equations"
StochasticDiffEq.WangLi3SMil_C — Type
WangLi3SMil_C()

WangLi3SMil_C: 3-Stage Milstein Method C (Nonstiff)

Third variant in the WangLi 3-stage Milstein family.

Method Properties

  • Strong Order: 1.0
  • Weak Order: 1.0
  • Time stepping: Fixed step size
  • Noise types: Depends on tableau (typically diagonal/scalar)
  • SDE interpretation: Itô

References

  • Wang and Li, "Three-stage stochastic Runge-Kutta methods for stochastic differential equations"
StochasticDiffEq.WangLi3SMil_D — Type
WangLi3SMil_D()

WangLi3SMil_D: 3-Stage Milstein Method D (Nonstiff)

Fourth variant in the WangLi 3-stage Milstein family.

Method Properties

  • Strong Order: 1.0
  • Weak Order: 1.0
  • Time stepping: Fixed step size
  • Noise types: Depends on tableau (typically diagonal/scalar)
  • SDE interpretation: Itô

References

  • Wang and Li, "Three-stage stochastic Runge-Kutta methods for stochastic differential equations"
StochasticDiffEq.WangLi3SMil_E — Type
WangLi3SMil_E()

WangLi3SMil_E: 3-Stage Milstein Method E (Nonstiff)

Fifth variant in the WangLi 3-stage Milstein family.

Method Properties

  • Strong Order: 1.0
  • Weak Order: 1.0
  • Time stepping: Fixed step size
  • Noise types: Depends on tableau (typically diagonal/scalar)
  • SDE interpretation: Itô

References

  • Wang and Li, "Three-stage stochastic Runge-Kutta methods for stochastic differential equations"
StochasticDiffEq.WangLi3SMil_F — Type
WangLi3SMil_F()

WangLi3SMil_F: 3-Stage Milstein Method F (Nonstiff)

Sixth and final variant in the WangLi 3-stage Milstein family.

Method Properties

  • Strong Order: 1.0
  • Weak Order: 1.0
  • Time stepping: Fixed step size
  • Noise types: Depends on tableau (typically diagonal/scalar)
  • SDE interpretation: Itô

When to Use (WangLi Family)

  • Compare all variants (A-F) to find best performance for your problem
  • Fixed step applications where step size is predetermined
  • Benchmarking against adaptive methods
  • When Milstein accuracy is needed with explicit fixed steps

References

  • Wang and Li, "Three-stage stochastic Runge-Kutta methods for stochastic differential equations"

Understanding Commutative Noise

Commutativity Condition

Noise terms g₁, g₂, ..., gₘ are commutative if:

[gᵢ, gⱼ] = gᵢ(∂gⱼ/∂x) - gⱼ(∂gᵢ/∂x) = 0

for all pairs (i,j).

Examples of Commutative Noise:

  1. Additive noise: g(u,t) = σ(t) (independent of u)
  2. Scalar multiplicative: g(u,t) = σ(t)u (same u dependence)
  3. Diagonal with same function: gᵢ(u,t) = σᵢ(t)h(u)

Examples of Non-commutative Noise:

  1. Different multiplicative terms: g₁ = σ₁u₁, g₂ = σ₂u₂
  2. Cross-coupling: g₁ = σ₁₁u₁ + σ₁₂u₂, g₂ = σ₂₁u₁ + σ₂₂u₂

Method Selection Guide

For Commutative Noise:

  1. RKMilCommute - Adaptive, excellent general choice
  2. WangLi3SMil methods - Fixed step, when dt is predetermined

For Non-commutative Noise:

  1. RKMilGeneral - Handles general case with Lévy area
  2. Fall back to SRI/SRA methods - More robust for complex noise

For Uncertain Commutativity:

  • Test with RKMilCommute first
  • If results seem incorrect, switch to RKMilGeneral or SRI methods

Computational Advantages

Commutative case:

  • No Lévy area computation needed
  • Simpler stochastic integrals
  • Higher efficiency per step
  • Better scalability to high dimensions

Non-commutative case:

  • Requires Lévy area approximation
  • More expensive per step
  • Uses specialized algorithms (LevyArea.jl integration)

Performance Tips

  1. Verify commutativity before using specialized methods
  2. Use appropriate interpretation (Itô vs Stratonovich)
  3. Consider problem dimension - benefits increase with system size
  4. Test accuracy - commutative methods may be more sensitive

Iterated Integrals and Lévy Area

For Commutative Noise (IICommutative):

Only simple stochastic integrals ∫₀ᵗ dWₛ are needed.

For Non-commutative Noise (IILevyArea):

Requires double integrals ∫₀ᵗ ∫₀ˢ dWᵤdWₛ (Lévy area).

RKMilGeneral automatically chooses optimal Lévy area algorithms via LevyArea.jl.

References

  • Kloeden, P.E., Platen, E., "Numerical Solution of Stochastic Differential Equations"
  • Wang and Li, "Three-stage stochastic Runge-Kutta methods for stochastic differential equations"
  • Kastner, F. and Rößler, A., "LevyArea.jl" for Lévy area computation