In [2]:
import numpy as np
%matplotlib inline
import matplotlib.pyplot as plt
import seaborn as sns
In [3]:
import antipackage
import github.ellisonbg.misc.vizarray as va
Downloading: https://raw.githubusercontent.com/ellisonbg/misc/master/vizarray.py
Using existing version: github.ellisonbg.misc.vizarray
Here is a function that produces standard Brownian motion using NumPy. This is also known as a Wiener Process.
In [4]:
def brownian(maxt, n):
"""Return one realization of a Brownian (Wiener) process with n steps and a max time of t."""
t = np.linspace(0.0,maxt,n)
h = t[1]-t[0]
Z = np.random.normal(0.0,1.0,n-1)
dW = np.sqrt(h)*Z
W = np.zeros(n)
W[1:] = dW.cumsum()
return t, W
Call the brownian function to simulate a Wiener process with 1000 steps and max time of 1.0. Save the results as two arrays t and W.
In [5]:
# YOUR CODE HERE
t, W = brownian(1.0, 1000)
In [6]:
assert isinstance(t, np.ndarray)
assert isinstance(W, np.ndarray)
assert t.dtype==np.dtype(float)
assert W.dtype==np.dtype(float)
assert len(t)==len(W)==1000
Visualize the process using plt.plot with t on the x-axis and W(t) on the y-axis. Label your x and y axes.
In [7]:
# YOUR CODE HERE
plt.plot(t,W)
plt.xlabel("W")
plt.ylabel("t")
Out[7]:
<matplotlib.text.Text at 0x7f94d3813d50>
In [8]:
assert True # this is for grading
Use np.diff to compute the changes at each step of the motion, dW, and then compute the mean and standard deviation of those differences.
In [11]:
# YOUR CODE HERE
dW = np.diff(W)
print dW
print np.mean(dW)
print np.std(dW)
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In [ ]:
assert len(dW)==len(W)-1
assert dW.dtype==np.dtype(float)
Write a function that takes $W(t)$ and converts it to geometric Brownian motion using the equation:
$$ X(t) = X_0 e^{((\mu - \sigma^2/2)t + \sigma W(t))} $$Use Numpy ufuncs and no loops in your function.
In [27]:
def geo_brownian(t, W, X0, mu, sigma):
X = X0*np.exp(((mu - sigma**2)/2.0)*t + sigma*W)
return X
In [28]:
assert True # leave this for grading
Use your function to simulate geometric brownian motion, $X(t)$ for $X_0=1.0$, $\mu=0.5$ and $\sigma=0.3$ with the Wiener process you computed above.
Visualize the process using plt.plot with t on the x-axis and X(t) on the y-axis. Label your x and y axes.
In [29]:
# YOUR CODE HERE
plt.plot(t,geo_brownian(t, W, 1.0, 0.5, 0.3))
plt.xlabel("X")
plt.ylabel("t")
Out[29]:
<matplotlib.text.Text at 0x7f94d375c050>
In [30]:
assert True # leave this for grading
In [ ]:
Content source: SJSlavin/phys202-2015-work
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