# -*- coding: utf-8 -*-
"""
Created on Mon Oct 17 16:18:35 2016

@author: florent
In this program, we set f(t)=sin(pi t) and compute
the Ito processes X,Y,Z starting at zero such that (for B a Browian motion)
dX=f(t)dt+tdB
dY=XdB
dZ=Y^2dB

We plot f,B,X,Y,Z
"""


import numpy as np
import numpy.random as npr
import scipy.stats as sps
import matplotlib.pyplot as plt

T=4
N=int(1e4)
deltat=T/float(N)

dB=np.sqrt(deltat)*npr.randn(N)
B=np.cumsum(dB)

t=np.linspace(start=0,stop=T,num=N)

dd=np.sin(np.pi*t)

dX=dd*deltat+t*dB
X=np.cumsum(dX)

dY=X*dB
Y=np.cumsum(dY)

dZ=(Y**2)*dB
Z=np.cumsum(dZ)

plt.plot(t,dd,color="m",label="f(t)")
plt.plot(t,B,color="r",label="B")
plt.plot(t,X,color="g",label="X")
plt.plot(t,Y,color="b",label="Y")
plt.plot(t,Z,color="y",label="Z")
plt.legend(loc="best")