SDE Solvers

# SDE Solvers

## Recommended Methods

For most Ito diagonal and scalar noise problems where a good amount of accuracy is required and mild stiffness may be an issue, the SRIW1 algorithm should do well. If the problem has additive noise, then SRA1 will be the optimal algorithm. For commutative noise, RKMilCommute is a strong order 1.0 method which utilizes the commutivity property to greatly speed up the Wiktorsson approximation and can choose between Ito and Stratonovich. For non-commutative noise, EM and EulerHeun are the choices (for Ito and Stratonovich interpretations respectively).

For stiff problems with diagonal noise, ImplicitRKMil is the most efficient method and can choose between Ito and Stratonovich. If the noise is non-diagonal, ImplicitEM and ImplicitEulerHeun are for Ito and Stratonovich respectively. For each of these 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.

## Mass Matrices and Stochastic DAEs

The ImplicitRKMil, ImplicitEM, and ImplicitEulerHeun methods can solve stochastic equations with mass matrices (including stochastic DAEs written in mass matrix form) when either symplectic=true or theta=1.

## 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:

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

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 $g_i(t,u)=a_i u$.

## 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.

# 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. 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.†

• 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.†

• SRA - The strong Order 1.5 methods for additive Ito and Stratonovich SDEs due to Rossler. Default tableau is for SRA1. Can handle non-diagonal and scalar additive noise.

• SRI - The strong Order 1.5 methods for diagonal/scalar Ito SDEs due to Rossler. Default tableau is for SRIW1.

• SRIW1 - An optimized version of SRIW1. Strong Order 1.5 for diagonal/scalar Ito SDEs.†

• SRA1 - An optimized version of SRA1. Strong Order 1.5 for additive Ito and Stratonovich SDEs. Can handle non-diagonal and scalar additive noise.†

Example usage:

sol = solve(prob,SRIW1())

### 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.

### Stiff Methods

• ImplicitEM - An order 0.5 Ito 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=1/2 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.

• ImplicitEulerHeun - An order 0.5 Stratonovich 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=1/2 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.

• ImplicitRKMil - An order 1.0 implicit method. This is a theta method which defaults to theta=1/2 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.

These methods interpret the mass matrix equation as:

$Mu' = f(t,u)dt + Mg(t,u)dW_t$

i.e. with no mass matrix inversion applied to the g term. Thus these methods apply noise per dependent variable instead of on the combinations of the dependent variables and this is designed for phenomenological noise on the dependent variables (like multiplicative or additive noise)

## 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.

## 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.†

#### Notes

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