Expected value and variance of some stochastic processes · 3 Diagrams per Page

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Expected value and variance of some stochastic
processes
14 Oct 2016
As an exercise, I illustrate how expected value and variance of some stochastic
processes evolve in time.
Poisson process
Let (N
t )t∈[0,∞)
be a Poisson process with rate λ. Then E(N
t)
= λt
and
. Below several paths of a Poisson process are shown. These are
well contained within E(N ) plus and minus the three time standard deviation
Var(Nt ) = λt
t
3σ
, where σ
= √λt
from above.
The gure was produced by the following Python code
import numpy as np
class Poisson_process(object):
def __init__(self, rate, dt=0.01):
self.N = 0
self.t = 0
self.dt = dt
self.rate = rate
def sample_path(self, T, reset=None):
path = []
if reset != None:
self.N = reset
for t in np.arange(0,T,self.dt):
k = np.random.poisson(self.rate*self.dt)
if k>0:
self.N+=1
path.append(self.N)
return path
rate = 1.
dt = 0.01
pp = Poisson_process(rate, dt=dt)
T = 25
ts = np.arange(0,T,dt)
import matplotlib as mpl
mpl.use('Agg')
import pylab as pl
from matplotlib import rc
rc('text', usetex=True)
pl.rcParams['text.latex.preamble'] = [
r'\usepackage{tgheros}',
# helvetica font
r'\usepackage{sansmath}',
# math-font matching helvetica
r'\sansmath'
# actually tell tex to use it!
r'\usepackage{siunitx}',
# micro symbols
r'\sisetup{detect-all}',
# force siunitx to use the fonts
]
fig = pl.figure()
fig.set_size_inches(4,3)
pl.plot(ts,ts, 'k', label=r'$\mu$')
pl.plot(ts, ts+3*np.sqrt(rate*ts), 'r', lw = 1., label=r'$\mu \p
m 3 \sigma$')
pl.plot(ts, ts-3*np.sqrt(rate*ts), 'r', lw = 1.)
for k in range(12):
path = pp.sample_path(T, reset=0)
pl.plot(ts, path, 'gray')
pl.legend(loc='upper left', frameon=False, prop={'size':12})
pl.xlabel('time')
pl.savefig("pp_sample.png", dpi=300, bbox_inches='tight')
Wiener process
Let (X
t )t∈[0,∞)
be a Wiener process. It is E(X
t)
= 0
and Var(X
t)
= t
. Below 15
sampled paths of the Wiener process are shown. It's trace are well contained
within 2σ or 3σ, where σ = √t is the standard deviation.
The gure was generated by the following Python code
import numpy as np
class Wiener_process(object):
def __init__(self, dt=0.01):
self.X = 0
self.t = 0
self.dt = dt
def sample_path(self, T, reset=None):
path = []
if reset != None:
self.X = reset
for t in np.arange(0,T,self.dt):
x = np.random.normal(0,np.sqrt(self.dt))
self.X+=x
path.append(self.X)
return path
dt = 0.01
wp = Wiener_process(dt=dt)
T = 50
ts = np.arange(0,T,dt)
import matplotlib as mpl
mpl.use('Agg')
import pylab as pl
from matplotlib import rc
rc('text', usetex=True)
pl.rcParams['text.latex.preamble'] = [
r'\usepackage{tgheros}',
# helvetica font
r'\usepackage{sansmath}',
# math-font matching helvetica
r'\sansmath'
# actually tell tex to use it!
r'\usepackage{siunitx}',
# micro symbols
r'\sisetup{detect-all}',
# force siunitx to use the fonts
]
fig = pl.figure()
fig.set_size_inches(5,3)
for k in range(15):
path = wp.sample_path(T, reset=0)
pl.plot(ts, path, 'gray')
pl.plot(ts, 2*np.sqrt(ts), 'r', linestyle='dashed', lw=1., label
=r'$2 \sigma$')
pl.plot(ts, -2*np.sqrt(ts), 'r', linestyle='dashed', lw=1.)
pl.plot(ts, 3*np.sqrt(ts), 'r', lw = 1., label=r'$3 \sigma$')
pl.plot(ts, -3*np.sqrt(ts), 'r', lw = 1.)
pl.ylim(-22.5,22.5)
pl.legend(loc='upper left', frameon=False, prop={'size':12})
pl.xlabel('time')
pl.savefig("wp_sample.png", dpi=300, bbox_inches='tight')
Ornstein-Uhlenbeck process
Let (X
t )t∈[0,∞)
be an Ornstein-Uhlenbeck process, that is a process de ned by
the stochastic differential equation
dXt = θ(μou − Xt )dt + σou dWt ,
with X
0
= a
. The expected of value this process is
E(Xt ) = ae
−θt
+ μou (1 − e
−θt
)
and the variance is
2
Var(Xt ) =
σou
2θ
(1 − e
−2θt
).
import numpy as np
class OU_process(object):
def __init__(self,X_0, mu, theta, sigma, dt=0.01):
self.X = X_0
self.dt = dt
self.mu = mu
self.theta = theta
self.sigma = sigma
self.setup()
def setup(self):
self.emdt = np.exp(-self.theta*self.dt)
self.a = np.sqrt(self.sigma**2/(2*self.theta) *
(1 - np.exp(-2*self.theta*self.dt)))
def step(self):
X_dt = self.X*self.emdt+self.mu*(1 - self.emdt) + \
self.a*np.random.normal()
self.X = X_dt
def path(self, T, reset=None):
if reset != None:
self.X = reset
path = []
for t in np.arange(0,T,self.dt):
path.append(self.X)
self.step()
return path
dt = 0.01
X_0 = -2
mu = 1.
theta = 1.
sigma = 1.
OU = OU_process(X_0, mu, theta, sigma, dt)
import matplotlib as mpl
mpl.use('Agg')
import pylab as pl
from matplotlib import rc
rc('text', usetex=True)
pl.rcParams['text.latex.preamble'] = [
r'\usepackage{tgheros}',
# helvetica font
r'\usepackage{sansmath}',
# math-font matching helvetica
r'\sansmath'
# actually tell tex to use it!
r'\usepackage{siunitx}',
# micro symbols
r'\sisetup{detect-all}',
# force siunitx to use the fonts
]
T = 10
ts = np.arange(0,T,dt)
mean = np.array([X_0*np.exp(-theta*t) + \
mu*(1-np.exp(-theta*t)) for t in ts])
var = np.array([sigma**2/(2*theta)* \
(1-np.exp(-2*theta*t)) for t in ts])
fig = pl.figure()
fig.set_size_inches(5,3)
for k in range(5):
path = OU.path(T, reset=X_0)
pl.plot(ts, path, 'gray')
pl.plot(ts, mean+2*np.sqrt(var), 'r', linestyle='dashed',
lw=1., label=r'$\mu \pm 2 \sigma$')
pl.plot(ts, mean-2*np.sqrt(var), 'r', linestyle='dashed',
lw=1.)
pl.ylim(-3,3)
pl.legend(loc='lower right', frameon=False, prop={'size':12})
pl.savefig("ou_sample.png", dpi=300, bbox_inches='tight')