PyS3DE 1.0.4

Python solvers for Stochastic Differential Equations

#Introduction

<h2>PyS³DE= "Py"thon Solver via "S"ympy + "S"ciPy/NumPy for "S"tochastic "D"ifferential "E"quations!</h2>


**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 with and without jumps.
<br>

<h2>Requirements:</h2>

1. <a href="http://sympy.org">Sympy</br>
2. <a href="http://www.scipy.org">Scipy and Numpy</a></br>
3. <a href="http://matplotlib.org">matplotlib</a>


<b>Note:</b> 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.

##Optional:

3. <a href="http://code.google.com/p/scitools/">scitools</a><br>

##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

## Demos

the files in demos include

- DiffusionJumps-1.sws DiffusionJumps.sws (for Sage notebook)
- demo.py
- sdedemo.tm, sdeJump.tm (for TeXmacs and Python plugin)
- sdedemo.ipynb (for IPython notebook)
- sdedemo.pdf



##Usages

* Symbolic Computation
<pre>
from sympy import *
from pysde 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)
</pre>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Got


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<i>t</i>+1)² <big><big><big>(</big></big></big>
-<i>a</i><i>t</i>(<i>t</i>₀ + 1)²+ <i>a</i> <i>t</i>₀(<i>t</i>₀+ 1)²+ <i>a</i><i>w</i>(<i>t</i>₀ + 1)²+ <i>x</i>₀<big><big><big>)</big></big></big><br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;───────────────────────────<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;
(<i>t</i>₀ + 1)²

* Numeric Computation
<pre>
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')
</pre>

##Note

* Symbolic/Numberic SDE solvers depend on part of the Scitools module which had been extracted and incorporated with the library,
which which it is not necessary to install scitools again.

* Schemes for simulating SDE’s with jumps adds, see demo.

##DEMO

1. python code: http://diffusion.cgu.edu.tw/ftp/sde (demo.py)
2. TeXmacs with Python plugin: http://diffusion.cgu.edu.tw/ftp/sde (sdedemo.py)
3. TeXmacs with Python plugin: SDE with jumps http://diffusion.cgu.edu.tw/ftp/sde (sdejump.py)
4. Sage notebook(): http://diffusion.cgu.edu.tw/ftp/sde DiffusionJumps.sws, DiffusionJumps-1.sws

##Developer:


chu-ching huang: [e-mail]