High Weak Order Methods

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

DRI1()

DRI1: Debrabant-Rößler Implicit Method (High Weak Order)

Adaptive high-order method optimized for weak convergence with minimized error constants. Excellent for Monte Carlo simulations and moment calculations.

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: All forms (diagonal, non-diagonal, non-commuting, scalar additive noise)
• SDE interpretation: Itô

When to Use

• Optimal for weak convergence requirements
• Monte Carlo simulations where statistical properties matter most
• Computing expectations, moments, and probability distributions
• When weak accuracy is more important than pathwise accuracy
• For problems requiring diverse noise types

Weak vs Strong Convergence

• Weak convergence: Convergence of expectations E[f(X_T)]
• Strong convergence: Pathwise convergence |XT - XT^h|
• DRI1 prioritizes weak convergence with optimized error constants

Algorithm Features

• Minimized error constants for better practical performance
• Handles complex noise structures including non-commuting terms
• Adaptive time stepping for efficiency

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)

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

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)

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

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)

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)

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)

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)

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)

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)

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)

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

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)

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

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

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)

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

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)

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)

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

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.

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)

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)

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

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)

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)

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)

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)