# SDE Solvers

For most Ito diagonal and scalar noise problems where a good amount of accuracy is required and mild stiffness may be an issue, the SOSRI algorithm should do well. If the problem has additive noise, then SOSRA will be the optimal algorithm. At low tolerances (<1e-4?) SRA3 will be more efficient, though SOSRA is more robust to stiffness. For commutative noise, RKMilCommute is a strong order 1.0 method which utilizes the commutivity property to greatly speed up the stochastic iterated integral approximation and can choose between Ito and Stratonovich. For non-commutative noise, difficult problems usually require adaptive time stepping in order to be efficient. In this case, LambaEM and LambaEulerHeun are adaptive and handle general non-diagonal problems (for Ito and Stratonovich interpretations respectively). If adaptivity isn't necessary, the EM and EulerHeun are good choices (for Ito and Stratonovich interpretations respectively).

For stiff problems with additive noise, the high order adaptive method SKenCarp is highly preferred and will solve problems with similar efficiency as ODEs. If possible, stiff problems should be converted to make use of this additive noise solver. If the noise term is large/stiff, then the split-step methods are required in order for the implicit methods to be stable. For Ito in this case, use ISSEM and for Stratonovich use ISSEulerHeun. These two methods can handle any noise form.

If the noise term is not too large, for stiff problems with diagonal noise, ImplicitRKMil is the most efficient method and can choose between Ito and Stratonovich. For each of the theta methods, the parameter theta can be chosen. The default is theta=1/2 which will not dampen numerical oscillations and thus is symmetric (and almost symplectic) and will lead to less error when noise is sufficiently small. However, theta=1/2 is not L-stable in the drift term, and thus one can receive more stability (L-stability in the drift term) with theta=1, but with a tradeoff of error efficiency in the low noise case. In addition, the option symplectic=true will turns these methods into an implicit Midpoint extension which is symplectic in distribution but has an accuracy tradeoff.

If only an estimation for the expected value of the solution is required, i.e., if one is only interested in an accurate draw from the distribution induced by a given SDE, the use of high weak order solvers is recommended. Specifically, DRI1 is preferred for a high number of Wiener processes. The weak stochastic Runge-Kutta solvers with weak order 2 due to Roessler are adaptive. All other high weak order solvers currently require a fixed step size.

## Special Noise Forms

Some solvers are for specialized forms of noise. Diagonal noise is the default setup. Non-diagonal noise is specified via setting noise_rate_prototype to a matrix in the SDEProblem type. A special form of non-diagonal noise, commutative noise, occurs when the noise satisfies the following condition:

$$$\sum_{i=1}^d g_{i,j_1}(t,x) \frac{\partial g_{k,j_2}(t,x)}{\partial x_i} = \sum_{i=1}^d g_{i,j_2}(t,x) \frac{\partial g_{k,j_1}(t,x)}{\partial x_i}$$$

for every $j_1,j_2$ and $k$. Additive noise is when $g(t,u)=g(t)$, i.e. is independent of u. Multiplicative noise is $g_i(t,u)=a_i u$.

## Iterated Integral Approximations

The difficulty of higher strong order integrators stems from the presence of iterated stochastic integrals

$$$I(h) = \int_0^h\int_0^sdW^1_tdW^2_s$$$

in these schemes.

The approximation of these iterated integrals can be avoided, if the diffusion matrix satisfies the special commutativity condition given above. Because of this, many methods are only applicable to problems that satisfy the commutativity condition. In other words, many methods can only handle specific noise cases, like diagonal noise or commutative noise, because of how this iterated integral approximation is computed.

However, the methods for general SDEs, like RKMilGeneral, perform a direct approximation of the iterated integrals. For those methods, the algorithms have an ii_approx keyword argument that allows one to specify the method for the approximation. The choices are:

• IICommutative: a simplification of the integral which assumes the noise commutativity property. If used on a non-commutative noise problem this will limit the strong convergence to 0.5.
• IILevyArea: computes the iterated integrals based on an approximation of the Levy area using the LevyArea.jl package: Kastner, F. and Rößler, A., arXiv: 2201.08424 Kastner, F. and Rößler, A., LevyArea.jl, 10.5281/ZENODO.5883748. The package supports the schemes: Fourier(), Milstein(), Wiktorsson(),MronRoe(). The optimal algorithm is automatically selected based on the dimension of the Brownian process and the step size. By passing a specific scheme, e.g. ii_approx=Fourier() methods can be manually selected. One must be careful when using the Levy area approximations in conjunction with adaptivity (adaptive=true) because the Levy area approximations draw random numbers that do not reflect the random numbers taken in a previous rejected step. This leads to a bias that increases with an increasing number of rejected steps.

Example: RKMilGeneral(;ii_approx=IILevyArea()).

## Special Keyword Arguments

• save_noise: Determines whether the values of W are saved whenever the timeseries is saved. Defaults to true.
• delta: The delta adaptivity parameter for the natural error estimator. Determines the balance between drift and diffusion error. For more details, see the publication.
• seed: Sets the seed for the random number generator. This overrides any seed set in the SDEProblem.

## Full List of Methods

### StochasticDiffEq.jl

Each of the StochasticDiffEq.jl solvers come with a linear interpolation. Orders are given in terms of strong order.

#### Nonstiff Methods

• EM- The Euler-Maruyama method. Strong Order 0.5 in the Ito sense. Has an optional argument split=true for controlling step splitting. When splitting is enabled, the stability with large diffusion eigenvalues is improved. Can handle all forms of noise, including non-diagonal, scalar, and colored noise. Fixed time step only.†
• LambaEM- A modified Euler-Maruyama method with adaptive time stepping with an error estimator based on Lamba and Rackauckas. Has an optional argument split=true for controlling step splitting. When splitting is enabled, the stability with large diffusion eigenvalues is improved. Strong Order 0.5 in the Ito sense. Can handle all forms of noise, including non-diagonal, scalar, and colored noise.†
• EulerHeun - The Euler-Heun method. Strong Order 0.5 in the Stratonovich sense. Can handle all forms of noise, including non-diagonal, scalar, and colored noise. Fixed time step only.†
• LambaEulerHeun - A modified Euler-Heun method with adaptive time stepping with an error estimator based on Lamba due to Rackauckas. Strong order 0.5 in the Stratonovich sense. Can handle all forms of noise, including non-diagonal, scalar, and colored noise.†
• RKMil - An explicit Runge-Kutta discretization of the strong order 1.0 Milstein method. Defaults to solving the Ito problem, but RKMil(interpretation=:Stratonovich) makes it solve the Stratonovich problem. Only handles scalar and diagonal noise.†
• RKMilCommute - An explicit Runge-Kutta discretization of the strong order 1.0 Milstein method for commutative noise problems. Defaults to solving the Ito problem, but RKMilCommute(interpretation=:Stratonovich) makes it solve the Stratonovich problem. Uses a 1.5/2.0 error estimate for adaptive time stepping.†
• RKMilGeneral(;interpretation=:Ito, ii_approx=IILevyArea() - An explicit Runge-Kutta discretization of the strong order 1.0 Milstein method for general non-commutative noise problems. Allows for a choice of interpretation between :Ito and :Stratonovich. Allows for a choice of iterated integral approximation.
• WangLi3SMil_A - fixed step-size explicit 3-stage Milstein methods for Ito problem with strong and weak order 1.0
• WangLi3SMil_B - fixed step-size explicit 3-stage Milstein methods for Ito problem with strong and weak order 1.0
• WangLi3SMil_C - fixed step-size explicit 3-stage Milstein methods for Ito problem with strong and weak order 1.0
• WangLi3SMil_D - fixed step-size explicit 3-stage Milstein methods for Ito problem with strong and weak order 1.0
• WangLi3SMil_E - fixed step-size explicit 3-stage Milstein methods for Ito problem with strong and weak order 1.0
• WangLi3SMil_F - fixed step-size explicit 3-stage Milstein methods for Ito problem with strong and weak order 1.0
• SRA - Adaptive strong order 1.5 methods for additive Ito and Stratonovich SDEs. Default tableau is for SRA1. Can handle diagonal, non-diagonal and scalar additive noise.
• SRI - Adaptive strong order 1.5 methods for diagonal/scalar Ito SDEs. Default tableau is for SRIW1.
• SRIW1 - Adaptive strong order 1.5 and weak order 2.0 for diagonal/scalar Ito SDEs.†
• SRIW2 - Adaptive strong order 1.5 and weak order 3.0 for diagonal/scalar Ito SDEs.†
• SOSRI - Stability-optimized adaptive strong order 1.5 and weak order 2.0 for diagonal/scalar Ito SDEs. Stable at high tolerances and robust to stiffness.†
• SOSRI2 - Stability-optimized adaptive strong order 1.5 and weak order 2.0 for diagonal/scalar Ito SDEs. Stable at high tolerances and robust to stiffness.†
• SRA1 - Adaptive strong order 1.5 for additive Ito and Stratonovich SDEs with weak order 2. Can handle diagonal, non-diagonal, and scalar additive noise.†
• SRA2 - Adaptive strong order 1.5 for additive Ito and Stratonovich SDEs with weak order 2. Can handle diagonal, non-diagonal, and scalar additive noise.†
• SRA3 - Adaptive strong order 1.5 for additive Ito and Stratonovich SDEs with weak order 3. Can handle non-diagonal and scalar additive noise.†
• SOSRA - A stability-optimized adaptive SRA. Strong order 1.5 for additive Ito and Stratonovich SDEs with weak order 2. Can handle diagonal, non-diagonal, and scalar additive noise. Stable at high tolerances and robust to stiffness.†
• SOSRA2 - A stability-optimized adaptive SRA. Strong order 1.5 for additive Ito and Stratonovich SDEs with weak order 2. Can handle diagonal, non-diagonal, and scalar additive noise. Stable at high tolerances and robust to stiffness.†

Example usage:

sol = solve(prob,SRIW1())

3-stage Milstein Methods WangLi3SMil_A, WangLi3SMil_B, WangLi3SMil_D, WangLi3SMil_E and WangLi3SMil_F are currently implemented for 1-dimensional and diagonal noise only.

#### Tableau Controls

For SRA and SRI, the following option is allowed:

• tableau: The tableau for an :SRA or :SRI algorithm. Defaults to SRIW1 or SRA1.

#### S-ROCK Methods

• SROCK1 - is a fixed step size stabilized explicit method for stiff problems. Defaults to solving th Ito problem but SROCK1(interpretation=:Stratonovich) can make it solve the Stratonovich problem. Strong order of convergence is 0.5 and weak order 1, but is optimised to get order 1 in case os scalar/diagonal noise.
• SROCKEM - is fixed step Euler-Mayurama with first order ROCK stabilization thus can handle stiff problems. Only for Ito problems. Defaults to strong and weak order 1.0, but can solve with weak order 0.5 as SROCKEM(strong_order_1=false). This method can handle 1-dimensional, diagonal and multi-dimensional noise.
• SROCK2 - is a weak second order and strong first order fixed step stabilized method for stiff Ito problems.This method can handle 1-dimensional, diagonal and multi-dimensional noise.
• SKSROCK - is fixed step stabilized explicit method for stiff Ito problems. Strong order 0.5 and weak order 1. This method has a better stability domain then SROCK1. Also it allows special post-processing techniques in case of ergodic dynamical systems, in the context of ergodic Brownian dynamics, to achieve order 2 accuracy. SKSROCK(;post_processing=true) will make use of post processing. By default it doesn't use post processing. Post processing is optional and under development. The rest of the method is completely functional and can handle 1-dimensional, diagonal and multi-dimensional noise.
• TangXiaoSROCK2 - is a fixed step size stabilized expicit method for stiff problems. Only for Ito problems. Weak order of 2 and strog order of 1. Has 5 versions with different stability domains which can be used as TangXiaoSROCK2(version_num=i) where i is 1-5. Under Development.

#### Stiff Methods

• ImplicitEM - An order 0.5 Ito drift-implicit method. This is a theta method which defaults to theta=1 or the Trapezoid method on the drift term. This method defaults to symplectic=false, but when true and theta=1/2 this is the implicit Midpoint method on the drift term and is symplectic in distribution. Can handle all forms of noise, including non-diagonal, scalar, and colored noise. Uses a 1.0/1.5 heuristic for adaptive time stepping.
• STrapezoid - An alias for ImplicitEM with theta=1/2
• SImplicitMidpoint - An alias for ImplicitEM with theta=1/2 and symplectic=true
• ImplicitEulerHeun - An order 0.5 Stratonovich drift-implicit method. This is a theta method which defaults to theta=1/2 or the Trapezoid method on the drift term. This method defaults to symplectic=false, but when true and theta=1 this is the implicit Midpoint method on the drift term and is symplectic in distribution. Can handle all forms of noise, including non-diagonal, scalar, and colored noise. Uses a 1.0/1.5 heuristic for adaptive time stepping.
• ImplicitRKMil - An order 1.0 drift-implicit method. This is a theta method which defaults to theta=1 or the Trapezoid method on the drift term. Defaults to solving the Ito problem, but ImplicitRKMil(interpretation=:Stratonovich) makes it solve the Stratonovich problem. This method defaults to symplectic=false, but when true and theta=1/2 this is the implicit Midpoint method on the drift term and is symplectic in distribution. Handles diagonal and scalar noise. Uses a 1.5/2.0 heuristic for adaptive time stepping.
• ISSEM - An order 0.5 split-step Ito implicit method. It is fully implicit, meaning it can handle stiffness in the noise term. This is a theta method which defaults to theta=1 or the Trapezoid method on the drift term. This method defaults to symplectic=false, but when true and theta=1/2 this is the implicit Midpoint method on the drift term and is symplectic in distribution. Can handle all forms of noise, including non-diagonal, scalar, and colored noise. Uses a 1.0/1.5 heuristic for adaptive time stepping.
• ISSEulerHeun - An order 0.5 split-step Stratonovich implicit method. It is fully implicit, meaning it can handle stiffness in the noise term. This is a theta method which defaults to theta=1 or the Trapezoid method on the drift term. This method defaults to symplectic=false, but when true and theta=1/2 this is the implicit Midpoint method on the drift term and is symplectic in distribution. Can handle all forms of noise, including non-diagonal,Q scalar, and colored noise. Uses a 1.0/1.5 heuristic for adaptive time stepping.
• SKenCarp - Adaptive L-stable drift-implicit strong order 1.5 for additive Ito and Stratonovich SDEs with weak order 2. Can handle diagonal, non-diagonal and scalar additive noise.*†

#### Derivative-Based Methods

The following methods require analytic derivatives of the diffusion term.

• PCEuler - The predictor corrector euler method. Strong Order 0.5 in the Ito sense. Requires the ggprime function, which is defined as

$$$\text{ggprime}^k(t,x) = \sum_{j=1}^m \sum_{i=1}^d g_{i,j}(t,x) \frac{\partial g_{k,j}(t,x)}{\partial x_i}.$$$

This can also be understood more intuitively in vector/matrix form as,

$$$\text{ggprime}(t,x) = \sum_{j=1}^m \bar{\mathcal{J}}[\vec g^{(j)}(t,x)] \vec g^{(j)}(t,x).$$$

where $\vec g^{(j)}$ is the noise vector for the j'th noise channel and $\bar{\mathcal{J}}$ is the Jacobian of the j'th noise vector.

The default settings for the drift implicitness is theta=0.5 and the diffusion implicitness is eta=0.5.

#### High Weak Order Methods

Note that none of the following methods are adaptive.

• SimplifiedEM - A simplified Euler-Maruyama method with weak order 1.0 and fixed step size. Can handle all forms of noise, including non-diagonal, scalar, and colored noise.†
• DRI1 - Adaptive step weak order 2.0 for Ito SDEs with minimized error constants (deterministic order 3). Can handle diagonal, non-diagonal, non-commuting, and scalar additive noise.†
• DRI1NM - Adaptive step weak order 2.0 for Ito SDEs with minimized error constants (deterministic order 3). Can handle non-mixing diagonal (i.e., du[k] = f(u[k])) and scalar additive noise.†
• RI1 - Adaptive step weak order 2.0 for Ito SDEs (deterministic order 3). Can handle diagonal, non-diagonal, non-commuting, and scalar additive noise.†
• RI3 - Adaptive step weak order 2.0 for Ito SDEs (deterministic order 3). Can handle diagonal, non-diagonal, non-commuting, and scalar additive noise.†
• RI5 - Adaptive step weak order 2.0 for Ito SDEs (deterministic order 3). Can handle diagonal, non-diagonal, non-commuting, and scalar additive noise.†
• RI6 - Adaptive step weak order 2.0 for Ito SDEs (deterministic order 2). Can handle diagonal, non-diagonal, non-commuting, and scalar additive noise.†
• RDI1WM - Fixed step weak order 1.0 for Ito SDEs (deterministic order 2). Can handle diagonal, non-diagonal, non-commuting, and scalar additive noise.†
• RDI2WM - Adaptive step weak order 2.0 for Ito SDEs (deterministic order 2). Can handle diagonal, non-diagonal, non-commuting, and scalar additive noise.†
• RDI3WM - Adaptive step weak order 2.0 for Ito SDEs (deterministic order 3). Can handle diagonal, non-diagonal, non-commuting, and scalar additive noise.†
• RDI4WM - Adaptive step weak order 2.0 for Ito SDEs (deterministic order 3). Can handle diagonal, non-diagonal, non-commuting, and scalar additive noise.†
• RS1 - Fixed step weak order 2.0 for Stratonovich SDEs (deterministic order 2). Can handle diagonal, non-diagonal, non-commuting, and scalar additive noise.†
• RS2 - Fixed step weak order 2.0 for Stratonovich SDEs (deterministic order 3). Can handle diagonal, non-diagonal, non-commuting, and scalar additive noise.†
• PL1WM - Fixed step weak order 2.0 for Ito SDEs (deterministic order 2). Can handle diagonal, non-diagonal, non-commuting, and scalar additive noise.†
• PL1WMA - Fixed step weak order 2.0 for Ito SDEs (deterministic order 2). Can handle additive noise.†
• NON - Fixed step weak order 2.0 for Stratonovich SDEs (deterministic order 4). Can handle diagonal, non-diagonal, non-commuting, and scalar additive noise.†
• SIEA - Fixed step weak order 2.0 for Ito SDEs (deterministic order 2). Can handle diagonal and scalar additive noise.† Stochastic generalization of the improved Euler method.
• SIEB - Fixed step weak order 2.0 for Ito SDEs (deterministic order 2). Can handle diagonal and scalar additive noise.† Stochastic generalization of the improved Euler method.
• SMEA - Fixed step weak order 2.0 for Ito SDEs (deterministic order 2). Can handle diagonal and scalar additive noise.† Stochastic generalization of the modified Euler method.
• SMEB - Fixed step weak order 2.0 for Ito SDEs (deterministic order 2). Can handle diagonal and scalar additive noise.† Stochastic generalization of the modified Euler method.

### StochasticCompositeAlgorithm

One unique feature of StochasticDiffEq.jl is the StochasticCompositeAlgorithm, which allows you to, with very minimal overhead, design a multimethod which switches between chosen algorithms as needed. The syntax is StochasticCompositeAlgorithm(algtup,choice_function) where algtup is a tuple of StochasticDiffEq.jl algorithms, and choice_function is a function which declares which method to use in the following step. For example, we can design a multimethod which uses EM() but switches to RKMil() whenever dt is too small:

choice_function(integrator) = (Int(integrator.dt<0.001) + 1)
alg_switch = StochasticCompositeAlgorithm((EM(),RKMil()),choice_function)

The choice_function takes in an integrator and thus all of the features available in the Integrator Interface can be used in the choice function.

### SimpleDiffEq.jl

This setup provides access to simplified versions of a few SDE solvers. They mostly exist for experimentation, but offer shorter compile times. They have limitations compared to StochasticDiffEq.jl.

• SimpleEM - A fixed timestep solve method for Euler-Maruyama. Only works with non-colored Gaussian noise.

Note that this setup is not automatically included with DifferentialEquaitons.jl. To use the following algorithms, you must install and use SimpleDiffEq.jl:

]add SimpleDiffEq
using SimpleDiffEq

### BridgeDiffEq.jl

Bridge.jl is a set of fixed timestep algorithms written in Julia. These methods are made and optimized for out-of-place functions on immutable (static vector) types. Note that this setup is not automatically included with DifferentialEquaitons.jl. To use the following algorithms, you must install and use BridgeDiffEq.jl:

Pkg.clone("https://github.com/JuliaDiffEq/BridgeDiffEq.jl")
using BridgeDiffEq
• BridgeEuler - Strong order 0.5 Euler-Maruyama method for Ito equations.†
• BridgeHeun - Strong order 0.5 Euler-Heun method for Stratonovich equations.†
• BridgeSRK - Strong order 1.0 derivative-free stochastic Runge-Kutta method for scalar (<:Number) Ito equations.†

†: Does not step to the interval endpoint. This can cause issues with discontinuity detection, and discrete variables need to be updated appropriately.

*: Note that although SKenCarp uses the same table as KenCarp3, solving a ODE problem using SKenCarp by setting g(du,u,p,t) = du .= 0 will take much more steps than KenCarp3 because error estimator of SKenCarp is different (because of noise terms) and default value of qmax (maximum permissible ratio of relaxing/tightening dt for adaptive steps) is smaller for StochasticDiffEq algorithms.