Weak SDE solver API

DRI1()

Adaptive Debrabant-Rößler method for weak approximation of Itô SDEs. DRI1 has weak order 2 and deterministic order 3 and supports scalar, diagonal, and non-diagonal noise.

IRI1(;
    autodiff = AutoForwardDiff(), concrete_jac = nothing, linsolve = nothing,
    nlsolve = NLNewton(), extrapolant = :constant, theta = 1,
    new_jac_conv_bound = 1.0e-3
)

Adaptive drift-implicit variant of the Rößler RI1 method for weak approximation of Itô SDEs. The nonlinear and linear solves can be customized with nlsolve and linsolve; autodiff controls Jacobian differentiation.

KomoriNON2

Butcher tableau for the second-order weak Komori NON2 method for Stratonovich SDEs.

COM()

COM: Commutative Stratonovich Method (High Weak Order)

Fixed step method optimized for commutative Stratonovich SDEs.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: Depends on implementation
• Time stepping: Fixed step size
• Noise types: Commutative noise only
• SDE interpretation: Stratonovich

When to Use

• Commutative Stratonovich SDEs
• When noise terms satisfy commutativity conditions
• More efficient alternative to NON for commutative cases
• Fixed step applications with commutative structure

Commutative Noise

Optimized for Stratonovich SDEs where:

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

for all noise terms.

Algorithm Features

• More efficient than NON for commutative cases
• Exploits commutativity for computational savings
• Specialized for Stratonovich interpretation

References

• Komori, Y., "Weak order stochastic Runge–Kutta methods for commutative stochastic differential equations", Journal of Computational and Applied Mathematics 203, pp. 57 – 79 (2007). Bibcode: 2007JCoAM.203...57K

DRI1NM()

DRI1NM: Debrabant-Rößler Implicit Non-Mixing Method (High Weak Order)

Specialized version of DRI1 for non-mixing diagonal and scalar additive noise problems.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 2.0 (optimized with minimized error constants)
• Deterministic Order: 3.0 (when noise = 0)
• Time stepping: Adaptive
• Noise types: Non-mixing diagonal and scalar additive noise
• SDE interpretation: Itô

When to Use

• Non-mixing diagonal problems: du[k] = f(u[k]) dt + σ[k] dW[k]
• Scalar additive noise problems
• When DRI1 is too general/expensive for the problem structure
• Monte Carlo simulations with special structure

Non-Mixing Diagonal Structure

Optimized for problems where:

du[1] = f₁(u[1])dt + σ₁ dW[1]
du[2] = f₂(u[2])dt + σ₂ dW[2]
...

Each component depends only on itself (no coupling).

Algorithm Advantages

• More efficient than general DRI1 for structured problems
• Exploits special structure for better performance
• Maintains weak order 2.0 with minimized constants

References

• Debrabant, K. and Rößler A., "Families of efficient second order Runge–Kutta methods for the weak approximation of Itô stochastic differential equations", Applied Numerical Mathematics 59, pp. 582–594 (2009). DOI: 10.1016/j.apnum.2008.03.012.

KomoriNON

Holds the Butcher tableau for Komori's NON method. (second weak order in Stratonovich sense)

NON()

NON: High Weak Order Method Fixed step weak order 2.0 for Stratonovich SDEs (deterministic order 4). Can handle diagonal, non-diagonal, non-commuting, and scalar additive noise.

References

• Komori, Y., Weak second-order stochastic Runge–Kutta methods for non-commutative stochastic differential equations, Journal of Computational and Applied Mathematics 206, pp. 158 – 173 (2007). DOI: 10.1016/j.cam.2006.06.006.

NON2()

NON2: Enhanced Non-commutative Stratonovich Method (High Weak Order)

Improved version of the NON method with enhanced efficiency for non-commutative Stratonovich SDEs.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 2.0
• Time stepping: Fixed step size
• Noise types: Non-commutative noise
• SDE interpretation: Stratonovich

When to Use

• Enhanced version of NON with better efficiency
• Non-commutative Stratonovich SDEs requiring improved performance
• When NON is too expensive or inefficient
• Modern alternative to classical NON method

Algorithm Features

• More efficient than original NON method
• Maintains weak order 2.0 convergence
• Enhanced computational techniques

References

• Komori, Y., & Burrage, K., "Supplement: Efficient weak second order stochastic Runge–Kutta methods for non-commutative Stratonovich stochastic differential equations", Journal of computational and applied mathematics, 235(17), pp. 5326-5329 (2011)

PL1WM()

PL1WM: Platen Weak Method 1 (High Weak Order)

Fixed step weak order 2.0 method from the classical Kloeden-Platen textbook.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 2.0
• Deterministic Order: 2.0 (when noise = 0)
• Time stepping: Fixed step size
• Noise types: All forms (diagonal, non-diagonal, non-commuting, scalar additive)
• SDE interpretation: Itô

When to Use

• Classical reference implementation for weak order 2.0
• Fixed step applications with predetermined step size
• Educational purposes and textbook examples
• Baseline comparison for more advanced methods

Algorithm Features

• Well-established classical method
• Simple implementation
• Standard reference from foundational SDE literature

References

• Kloeden, P.E., Platen, E., "Numerical Solution of Stochastic Differential Equations", Springer. Berlin Heidelberg (2011). ISBN 978-3-540-54062-5. DOI: 10.1007/978-3-662-12616-5.

PL1WMA()

PL1WMA: Platen Weak Method 1 Additive (High Weak Order)

Specialized version of PL1WM optimized for additive noise problems.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 2.0
• Deterministic Order: 2.0 (when noise = 0)
• Time stepping: Fixed step size
• Noise types: Additive noise only
• SDE interpretation: Itô

When to Use

• Additive noise problems with fixed step size
• When PL1WM is too general for additive structure
• Classical reference for additive noise weak methods
• Educational and benchmarking purposes

Additive Noise Structure

Specialized for SDEs of the form:

\[du = f(u,t)dt + σ(t) dW\]

where diffusion σ doesn't depend on solution u.

Algorithm Features

• More efficient than PL1WM for additive problems
• Classical foundation method
• Simplified implementation for additive case

References

• Kloeden, P.E., Platen, E., "Numerical Solution of Stochastic Differential Equations", Springer. Berlin Heidelberg (2011). ISBN 978-3-540-54062-5. DOI: 10.1007/978-3-662-12616-5.

RDI1WM()

RDI1WM: Runge-Kutta Debrabant Implicit 1 Weak Method (High Weak Order)

Fixed step method with weak order 1.0 for Itô SDEs.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 1.0
• Deterministic Order: 2.0 (when noise = 0)
• Time stepping: Fixed step size
• Noise types: All forms (diagonal, non-diagonal, non-commuting, scalar additive)
• SDE interpretation: Itô

When to Use

• Fixed step applications where step size is predetermined
• When weak order 1.0 is sufficient
• Simpler alternative to higher-order weak methods
• Baseline for comparing higher-order methods

References

• Debrabant, K. and Rößler A., "Classification of Stochastic Runge–Kutta Methods for the Weak Approximation of Stochastic Differential Equations", Mathematics and Computers in Simulation 77, pp. 408-420 (2008). DOI: 10.1016/j.matcom.2007.04.016

RDI2WM()

RDI2WM: Runge-Kutta Debrabant Implicit 2 Weak Method (High Weak Order)

Adaptive weak order 2.0 method for Itô SDEs with deterministic order 2.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 2.0
• Deterministic Order: 2.0 (when noise = 0)
• Time stepping: Adaptive
• Noise types: All forms (diagonal, non-diagonal, non-commuting, scalar additive)
• SDE interpretation: Itô

When to Use

• Weak order 2.0 problems with adaptive stepping
• Alternative to DRI1 and RI methods
• When deterministic order 2.0 is sufficient
• Monte Carlo simulations requiring adaptive control

References

• Debrabant, K. and Rößler A., "Classification of Stochastic Runge–Kutta Methods for the Weak Approximation of Stochastic Differential Equations", Mathematics and Computers in Simulation 77, pp. 408-420 (2008). DOI: 10.1016/j.matcom.2007.04.016.

RDI3WM()

RDI3WM: Runge-Kutta Debrabant Implicit 3 Weak Method (High Weak Order)

Adaptive weak order 2.0 method with higher deterministic order 3.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 2.0
• Deterministic Order: 3.0 (when noise = 0)
• Time stepping: Adaptive
• Noise types: All forms (diagonal, non-diagonal, non-commuting, scalar additive)
• SDE interpretation: Itô

When to Use

• When both weak order 2.0 and deterministic order 3.0 are needed
• Problems with significant deterministic components
• Alternative to DRI1 with different characteristics
• High accuracy requirements for both stochastic and deterministic parts

References

• Debrabant, K. and Rößler A., "Classification of Stochastic Runge–Kutta Methods for the Weak Approximation of Stochastic Differential Equations", Mathematics and Computers in Simulation 77, pp. 408-420 (2008). DOI: 10.1016/j.matcom.2007.04.016.

RDI4WM()

RDI4WM: Runge-Kutta Debrabant Implicit 4 Weak Method (High Weak Order)

Fourth variant in the RDI family with weak order 2.0 and deterministic order 3.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 2.0
• Deterministic Order: 3.0 (when noise = 0)
• Time stepping: Adaptive
• Noise types: All forms (diagonal, non-diagonal, non-commuting, scalar additive)
• SDE interpretation: Itô

When to Use

• Final alternative in the RDI family
• When other RDI methods don't provide desired performance
• Completing comprehensive RDI method comparisons
• Research applications requiring all RDI variants

References

• Debrabant, K. and Rößler A., "Classification of Stochastic Runge–Kutta Methods for the Weak Approximation of Stochastic Differential Equations", Mathematics and Computers in Simulation 77, pp. 408-420 (2008). DOI: 10.1016/j.matcom.2007.04.016.

RI1()

RI1: Rößler Implicit Method 1 (High Weak Order)

Adaptive weak order 2.0 method for Itô SDEs with deterministic order 3.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 2.0
• Deterministic Order: 3.0 (when noise = 0)
• Time stepping: Adaptive
• Noise types: All forms (diagonal, non-diagonal, non-commuting, scalar additive)
• SDE interpretation: Itô

When to Use

• General weak convergence problems
• Monte Carlo simulations with various noise structures
• When weak order 2.0 is sufficient
• Alternative to DRI1 with different characteristics

References

• Rößler A., "Second Order Runge–Kutta Methods for Itô Stochastic Differential Equations", SIAM J. Numer. Anal., 47, pp. 1713-1738 (2009). DOI: 10.1137/060673308.

RI3()

RI3: Rößler Implicit Method 3 (High Weak Order)

Alternative adaptive weak order 2.0 method with different stability characteristics.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 2.0
• Deterministic Order: 3.0 (when noise = 0)
• Time stepping: Adaptive
• Noise types: All forms (diagonal, non-diagonal, non-commuting, scalar additive)
• SDE interpretation: Itô

When to Use

• Alternative to RI1 with different characteristics
• When RI1 performance is unsatisfactory
• Benchmarking different weak order 2.0 methods

References

• Rößler A., "Second Order Runge–Kutta Methods for Itô Stochastic Differential Equations", SIAM J. Numer. Anal., 47, pp. 1713-1738 (2009). DOI: 10.1137/060673308

RI5()

RI5: Rößler Implicit Method 5 (High Weak Order)

Another variant in the RI family of weak order 2.0 methods.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 2.0
• Deterministic Order: 3.0 (when noise = 0)
• Time stepping: Adaptive
• Noise types: All forms (diagonal, non-diagonal, non-commuting, scalar additive)
• SDE interpretation: Itô

When to Use

• Part of RI family comparison studies
• When other RI methods don't provide desired characteristics
• Research applications requiring different RI variants

References

• Rößler A., "Second Order Runge–Kutta Methods for Itô Stochastic Differential Equations", SIAM J. Numer. Anal., 47, pp. 1713-1738 (2009). DOI: 10.1137/060673308

RI6()

RI6: Rößler Implicit Method 6 (High Weak Order)

Final method in the RI family with deterministic order 2 (lower than other RI methods).

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 2.0
• Deterministic Order: 2.0 (when noise = 0)
• Time stepping: Adaptive
• Noise types: All forms (diagonal, non-diagonal, non-commuting, scalar additive)
• SDE interpretation: Itô

When to Use

• When lower deterministic order is acceptable
• Potentially more efficient than RI1/RI3/RI5
• Completing RI family comparisons

Algorithm Features

• Lower deterministic order may reduce computational cost
• Still maintains weak order 2.0 for stochastic problems
• Final variant in the comprehensive RI family

References

• Rößler A., "Second Order Runge–Kutta Methods for Itô Stochastic Differential Equations", SIAM J. Numer. Anal., 47, pp. 1713-1738 (2009). DOI: 10.1137/060673308

RS1()

RS1: Rößler Stratonovich Method 1 (High Weak Order)

Fixed step weak order 2.0 method specifically designed for Stratonovich SDEs.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 2.0
• Deterministic Order: 2.0 (when noise = 0)
• Time stepping: Fixed step size
• Noise types: All forms (diagonal, non-diagonal, non-commuting, scalar additive)
• SDE interpretation: Stratonovich

When to Use

• Stratonovich SDEs requiring weak order 2.0
• Fixed step applications with predetermined step size
• Problems naturally formulated in Stratonovich interpretation
• When physical interpretation requires Stratonovich calculus

Stratonovich Interpretation

Optimized for SDEs in Stratonovich form:

\[du = f(u,t)dt + g(u,t)∘dW\]

where ∘ denotes Stratonovich integration.

References

• Rößler A., "Second order Runge–Kutta methods for Stratonovich stochastic differential equations", BIT Numerical Mathematics 47, pp. 657-680 (2007) DOI: 10.1007/s10543-007-0130-3.

RS2()

RS2: Rößler Stratonovich Method 2 (High Weak Order)

Alternative fixed step weak order 2.0 method for Stratonovich SDEs with higher deterministic order.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 2.0
• Deterministic Order: 3.0 (when noise = 0)
• Time stepping: Fixed step size
• Noise types: All forms (diagonal, non-diagonal, non-commuting, scalar additive)
• SDE interpretation: Stratonovich

When to Use

• Stratonovich SDEs with significant deterministic components
• When higher deterministic accuracy than RS1 is needed
• Fixed step applications requiring better deterministic performance
• Benchmarking against RS1

Algorithm Features

• Higher deterministic order than RS1
• May be more expensive per step than RS1
• Better for problems with large deterministic components

References

• Rößler A., "Second order Runge–Kutta methods for Stratonovich stochastic differential equations", BIT Numerical Mathematics 47, pp. 657-680 (2007). DOI: 10.1007/s10543-007-0130-3

RoesslerRI

Holds the Butcher tableaus for a Roessler RI method. (high weak order)

RoesslerRS

Holds the Butcher tableaus for a Roessler RS method. (high weak order in Stratonovich sense)

SIEA()

SIEA: Stochastic Improved Euler A Method (High Weak Order)

Stochastic generalization of the improved Euler method for Itô SDEs.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 2.0
• Deterministic Order: 2.0 (when noise = 0)
• Time stepping: Fixed step size
• Noise types: Diagonal and scalar additive noise
• SDE interpretation: Itô

When to Use

• Fixed step applications with diagonal/scalar additive noise
• When stochastic version of improved Euler is desired
• Educational purposes (connection to classical methods)
• Baseline for Tocino-Vigo-Aguiar method comparisons

Algorithm Features

• Based on classical improved Euler method
• Specialized for additive noise structures
• Simple and well-understood foundation

References

• Tocino, A. and Vigo-Aguiar, J., "Weak Second Order Conditions for Stochastic Runge-Kutta Methods", SIAM Journal on Scientific Computing 24, pp. 507-523 (2002). DOI: 10.1137/S1064827501387814.

SIEB()

SIEB: Stochastic Improved Euler B Method (High Weak Order)

Alternative stochastic generalization of the improved Euler method.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 2.0
• Deterministic Order: 2.0 (when noise = 0)
• Time stepping: Fixed step size
• Noise types: Diagonal and scalar additive noise
• SDE interpretation: Itô

When to Use

• Alternative to SIEA with different coefficients
• Fixed step applications requiring different stability properties
• Comparing different improved Euler generalizations
• When SIEA performance is unsatisfactory

Algorithm Features

• Variant B of stochastic improved Euler approach
• Different coefficients than SIEA
• May have different stability or accuracy characteristics

References

• Tocino, A. and Vigo-Aguiar, J., "Weak Second Order Conditions for Stochastic Runge-Kutta Methods", SIAM Journal on Scientific Computing 24, pp. 507-523 (2002). DOI: 10.1137/S1064827501387814.

SMEA()

SMEA: Stochastic Modified Euler A Method (High Weak Order)

Stochastic generalization of the modified Euler method for Itô SDEs.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 2.0
• Deterministic Order: 2.0 (when noise = 0)
• Time stepping: Fixed step size
• Noise types: Diagonal and scalar additive noise
• SDE interpretation: Itô

When to Use

• Fixed step applications with diagonal/scalar additive noise
• When stochastic version of modified Euler is desired
• Alternative to SIEA with different characteristics
• Educational and comparison purposes

Algorithm Features

• Based on classical modified Euler method
• Different approach than SIEA for same problem class
• Specialized for additive noise structures

References

• Tocino, A. and Vigo-Aguiar, J., "Weak Second Order Conditions for Stochastic Runge-Kutta Methods", SIAM Journal on Scientific Computing 24, pp. 507-523 (2002). DOI: 10.1137/S1064827501387814.

SMEB()

SMEB: Stochastic Modified Euler B Method (High Weak Order)

Alternative stochastic generalization of the modified Euler method.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 2.0
• Deterministic Order: 2.0 (when noise = 0)
• Time stepping: Fixed step size
• Noise types: Diagonal and scalar additive noise
• SDE interpretation: Itô

When to Use

• Alternative to SMEA with different coefficients
• Fixed step applications requiring different characteristics
• Completing Tocino-Vigo-Aguiar method family comparisons
• When SMEA performance is unsatisfactory

Algorithm Features

• Variant B of stochastic modified Euler approach
• Different coefficients than SMEA
• Completes the family of Tocino-Vigo-Aguiar methods

References

• Tocino, A. and Vigo-Aguiar, J., "Weak Second Order Conditions for Stochastic Runge-Kutta Methods", SIAM Journal on Scientific Computing 24, pp. 507-523 (2002). DOI: 10.1137/S1064827501387814

W2Ito1()

W2Ito1: Wang-Tang-Xiao Weak Order 2 Method (High Weak Order)

Efficient weak second-order method for Itô SDEs with adaptive stepping.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 2.0
• Deterministic Order: 3.0 (when noise = 0)
• Time stepping: Adaptive
• Noise types: All forms (diagonal, non-diagonal, non-commuting, scalar additive)
• SDE interpretation: Itô

When to Use

• Modern efficient weak order 2.0 method
• When computational efficiency is important for weak convergence
• Alternative to older weak order 2.0 methods
• Monte Carlo simulations requiring good performance

Algorithm Features

• Designed for computational efficiency
• Good balance of accuracy and cost for weak problems
• More recent development than classical methods

References

• Tang, X., & Xiao, A., "Efficient weak second-order stochastic Runge–Kutta methods for Itô stochastic differential equations", BIT Numerical Mathematics, 57, 241-260 (2017). DOI: 10.1007/s10543-016-0618-9

constructDRI1([T = Float64], [T2 = Float64])

Construct the RoesslerRI tableau used by DRI1, with coefficient element type T and abscissa element type T2.

constructRI1([T = Float64], [T2 = Float64])

Construct the RoesslerRI tableau used by RI1, with coefficient element type T and abscissa element type T2.

constructRI3([T = Float64], [T2 = Float64])

Construct the RoesslerRI tableau used by RI3, with coefficient element type T and abscissa element type T2.

constructRI5([T = Float64], [T2 = Float64])

Construct the RoesslerRI tableau used by RI5, with coefficient element type T and abscissa element type T2.

constructRI6([T = Float64], [T2 = Float64])

Construct the RoesslerRI tableau used by RI6, with coefficient element type T and abscissa element type T2.

constructRDI1WM([T = Float64], [T2 = Float64])

Construct the RoesslerRI tableau used by RDI1WM, with coefficient element type T and abscissa element type T2.

constructRDI2WM([T = Float64], [T2 = Float64])

Construct the RoesslerRI tableau used by RDI2WM, with coefficient element type T and abscissa element type T2.

constructRDI3WM([T = Float64], [T2 = Float64])

Construct the RoesslerRI tableau used by RDI3WM, with coefficient element type T and abscissa element type T2.

constructRDI4WM([T = Float64], [T2 = Float64])

Construct the RoesslerRI tableau used by RDI4WM, with coefficient element type T and abscissa element type T2.

constructRS1([T = Float64], [T2 = Float64])

Construct the RoesslerRS tableau used by RS1, with coefficient element type T and abscissa element type T2.

constructRS2([T = Float64], [T2 = Float64])

Construct the RoesslerRS tableau used by RS2, with coefficient element type T and abscissa element type T2.

constructNON([T = Float64])

Construct the KomoriNON tableau used by NON, with coefficient element type T.

constructNON2([T = Float64])

Construct the KomoriNON2 tableau used by NON2, with coefficient element type T.

checkRIOrder(tableau; tol = 1.0e-6, ps = 2)

Evaluate the weak order conditions of a RoesslerRI tableau through order ps. Returns one Boolean per condition, using tol for coefficient comparisons. Supported values of ps are 1 and 2.

checkRSOrder(tableau; tol = 1.0e-6, ps = 2)

Evaluate the weak order conditions of a RoesslerRS tableau through order ps. Returns one Boolean per condition, using tol for coefficient comparisons. Supported values of ps are 1 and 2.

checkNONOrder(tableau; tol = 1.0e-6)

Evaluate the weak and deterministic order conditions of a KomoriNON or KomoriNON2 tableau. Returns one Boolean per condition, using tol for coefficient comparisons.