# -*- coding: utf-8 -*-
"""
Created on Wed Oct 19 13:11:42 2016

@author: florent
In this program, we set f(t)=exp(-t) and compute
the Ito process X starting at zero such that (for B a Browian motion)
dX=f(t)dB


We plot f,B,X
"""

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

T=5.

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


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

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

expmt=np.exp(-t)
X=np.cumsum(expmt*dB)
plt.plot(t,expmt,"g--",label="f(t)")
plt.plot(t,B,color="r",label="B")
plt.plot(t,X,color="b",label="X")
plt.legend(loc="best")