Introduction (1.0.5)

PyS³DE= Python Solver via Sympy + SciPy/NumPy for Stochastic Differential Equations!

PyS³DE is a solver of stochastic differential equations (SDE) implemented by Python, which both symbolic and numeric schemems are supported. Numerical solvers include schemes for both w/o jumps.

Requirements:

1. Sympy
   
2. Scipy and Numpy
   
3. matplotlib

Note:

1. Scitools is used to generated the picture of simulated data only. To keep minimal necessary Python modules/libraries, it would be removed in TODO list. (Thanks, Yoav Ram) Necessary required submodules of scitools have been added to PySDE since 0.4; this means that scitools is not to be installed.

• after (includeing) 1.0.5, numpy.oldnumeric dependecy was removed ; in otherwords, PySDE works with any version of numpy. (Thanks to Lars Ericson erxnmedia@hotmail.com) ##Optional:

3. scitools
   

Installation

• Extract the source and enter the source directory

  • Python2 version: cd Python2; python setup.py install
  • Python3 version: cd Python3; python setup.py install

Usages

• Symbolic Computation

from sympy import *
from pysde.sde import *
""" Main Codes Here """
x,dx,w,dw,t,dt,a=symbols('x dx w dw t dt a')
x0 =Symbol('x0'); t0 = Symbol('t0')
drift=2\*x/(1+t)-a\*(1+t)\*\*2;diffusion=a\*(1+t)**2
sol=SDE_solver(drift,diffusion,t0,x0)
pprint(sol)  

       Got

            (t+1)² ( -at(t₀ + 1)²+ a t₀(t₀+ 1)²+ aw(t₀ + 1)²+ x₀)
            ───────────────────────────
                                            (t₀ + 1)²

• Numeric Computation

import matplotlib.pylab as plt
from matplotlib import rc
rc('font',\*\*{'family':'sans-serif','sans-serif':['Helvetica']})
rc('text', usetex=True)
""" setup picture info """
plt.figure(figsize=(5,2))
plt.ylim(-0.5,1.5)
""" Initial data """
x0=1.;t0=0.;tn=10.
x,dx=symbols('x dx')
[a,b,c,d]=[0,-1.,0,1.]
drift=a+b\*x
diffusion=c+d\*x#
nt=200
T= linspace(t0, tn, nt+1)
""" Numerical Computation"""
X=Euler(drift,diffusion,x0,t0,tn,nt)
X,Y=Milstein(drift,diffusion,x0,t0,tn,nt)
"""Make picture"""
plt.plot(T, X, color="blue", linewidth=2.5, linestyle="-", label="Euler")
plt.plot(T, Y, color="red", linewidth=2.5, linestyle="--", label="Milstein")
plt.plot(T, np.exp(-T), color="green", linewidth=2.5, linestyle="--", label=r"$\exp(-t)$")
plt.ylim(X.min()-0.2, X.max()+0.2)
plt.title(r"$d X_t=-dt+d W_t,X_0=1$")
plt.legend()
plt.savefig('Milstein.eps')

• SDE with jumps
  

Note

• Symbolic/Numeric SDE solvers depend on part of the Scitools module which had been extracted and incorporated with the library, which it is not necessary to install scitools again. (removed since 0.4)

• Schemes and examples for simulating SDE’s with jumps adds, see demo/sdedemo.ipynb.

DEMO

1. python code: demo/demo.py)

• TeXmacs with Python plugin: demo/sdedemo.tm, demo/sdeJump.tm
• Sage notebook(): demo/DiffusionJumps.sws, demo/DiffusionJumps-1.sws
• Jupyter/notebook: demo/sdedemo.ipynb

Developer:

1. chu-ching huang: cchuang2009@gmail.com

• Lars Ericson: erxnmedia@hotmail.com