Numerical integration of stochastic differential equations (SDE)
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!
These work with scalar or vector equations. They will choose an algorithm for you. Or you can use a specific algorithm directly:
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)
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:
- 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).
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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