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Numerical integration of stochastic differential equations (SDE)

## Project description

Numerical integration of Ito or Stratonovich SDEs.

## Overview

sdeint is a collection of numerical algorithms for integrating Ito and Stratonovich stochastic ordinary differential equations (SODEs). It has simple functions that can be used in a similar way to scipy.integrate.odeint() or MATLAB’s ode45.

There already exist some python and MATLAB packages providing Euler-Maruyama and Milstein algorithms, and a couple of others. So why am I bothering to make another package?

It is because there has been 25 years of further research with better methods but for some reason I can’t find any open source reference implementations. Not even for those methods published by Kloeden and Platen way back in 1992. So I will aim to gradually add some improved methods here.

This is prototype code in python, so not aiming for speed. Later can always rewrite these with loops in C when speed is needed.

Warning: this is an early pre-release. Wait for version 1.0. Bug reports are very welcome!

## functions

itoint(f, G, y0, tspan) for Ito equation dy = f(y,t)dt + G(y,t)dW
stratint(f, G, y0, tspan) for Stratonovich equation dy = f(y,t)dt + G(y,t)∘dW

These work with scalar or vector equations. They will choose an algorithm for you. Or you can use a specific algorithm directly:

## specific algorithms:

itoEuler(f, G, y0, tspan): the Euler-Maruyama algorithm for Ito equations.
stratHeun(f, G, y0, tspan): the Stratonovich Heun algorithm for Stratonovich equations.
itoSRI2(f, G, y0, tspan): the Rößler2010 order 1.0 strong Stochastic Runge-Kutta algorithm SRI2 for Ito equations.
itoSRI2(f, [g1,...,gm], y0, tspan): as above, with G matrix given as a separate function for each column (gives speedup for large m or complicated G).
stratSRS2(f, G, y0, tspan): the Rößler2010 order 1.0 strong Stochastic Runge-Kutta algorithm SRS2 for Stratonovich equations.
stratSRS2(f, [g1,...,gm], y0, tspan): as above, with G matrix given as a separate function for each column (gives speedup for large m or complicated G).
stratKP2iS(f, G, y0, tspan): the Kloeden and Platen two-step implicit order 1.0 strong algorithm for Stratonovich equations.
For more information and advanced options see the documentation for each function.

### utility functions:

deltaW(N, m, h): Generate increments of m independent Wiener processes for each of N time intervals of length h.
Repeated integrals by the method of Kloeden, Platen and Wright (1992):
Ikpw(dW, h, n=5): Approximate repeated Ito integrals.
Jkpw(dW, h, n=5): Approximate repeated Stratonovich integrals.
Repeated integrals by the method of Wiktorsson (2001):
Iwik(dW, h, n=5): Approximate repeated Ito integrals.
Jwik(dW, h, n=5): Approximate repeated Stratonovich integrals.

## Examples:

Integrate the one-dimensional Ito equation with initial condition x0 = 0.1
```import numpy as np
import sdeint

a = 1.0
b = 0.8
tspan = np.linspace(0.0, 5.0, 5001)
x0 = 0.1

def f(x, t):
return -(a + x*b**2)*(1 - x**2)

def g(x, t):
return b*(1 - x**2)

result = sdeint.itoint(f, g, x0, tspan)
```
Integrate the two-dimensional vector Ito equation where x = (x1, x2),   dW = (dW1, dW2) and with initial condition x0 = (3.0, 3.0)
```import numpy as np
import sdeint

A = np.array([[-0.5, -2.0],
[ 2.0, -1.0]])

B = np.diag([0.5, 0.5]) # diagonal, so independent driving Wiener processes

tspan = np.linspace(0.0, 10.0, 10001)
x0 = np.array([3.0, 3.0])

def f(x, t):
return A.dot(x)

def G(x, t):
return B

result = sdeint.itoint(f, G, x0, tspan)
```

## References for these algorithms:

itoEuler:
G. Maruyama (1955) Continuous Markov processes and stochastic equations
stratHeun:
W. Rumelin (1982) Numerical Treatment of Stochastic Differential Equations
R. Mannella (2002) Integration of Stochastic Differential Equations on a Computer
K. Burrage, P. M. Burrage and T. Tian (2004) Numerical methods for strong solutions of stochastic differential equations: an overview
itoSRI2, stratSRS2:
A. Rößler (2010) Runge-Kutta Methods for the Strong Approximation of Solutions of Stochastic Differential Equations
stratKP2iS:
P. Kloeden and E. Platen (1999) Numerical Solution of Stochastic Differential Equations, revised and updated 3rd printing
Ikpw, Jkpw:
P. Kloeden, E. Platen and I. Wright (1992) The approximation of multiple stochastic integrals
Iwik, Jwik:
M. Wiktorsson (2001) Joint Characteristic Function and Simultaneous Simulation of Iterated Ito Integrals for Multiple Independent Brownian Motions

## TODO

• Rewrite Iwik() and Jwik() so they don’t waste so much memory.
• Fix stratKP2iS(). In the unit tests it is currently less accurate than itoEuler() and this is likely due to a bug.
• Implement the Ito version of the Kloeden and Platen two-step implicit alogrithm.
• Add more strong stochastic Runge-Kutta algorithms. Perhaps starting with Burrage and Burrage (1996)
• Currently prioritizing those algorithms that work for very general d-dimensional systems with arbitrary noise coefficient matrix, and which are derivative free. Eventually will add special case algorithms that give a speed increase for systems with certain symmetries. That is, 1-dimensional systems, systems with scalar noise, diagonal noise or commutative noise, etc. The idea is that itoint() and stratint() will detect these situations and dispatch to the most suitable algorithm.
• Eventually implement the main loops in C for speed.
• Some time in the dim future, implement support for stochastic delay differential equations (SDDEs).

## See also:

nsim: Framework that uses this sdeint library to enable massive parallel simulations of SDE systems (using multiple CPUs or a cluster) and provides some tools to analyze the resulting timeseries. https://github.com/mattja/nsim

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