IlievskiV

def geometric_brownian_motion(G0, mu, sigma, N, T):
    """Simulates a Geometric Brownian Motion.
    
    :param float G0: initial value
    :param float mu: drift coefficient
    :param float sigma: diffusion coefficient
    :param int N: number of discrete steps
    :param int T: number of continuous time steps
    :return list: the geometric Brownian Motion
    """

IlievskiV

import numpy as np

def drifted_brownian_motion(mu, sigma, N, T, seed=42):
    """Simulates a Brownian Motion with drift.
    
    :param float mu: drift coefficient
    :param float sigma: volatility coefficient
    :param int N : number of discrete steps
    :param int T: number of continuous time steps
    :param int seed: initial seed of the random generator

IlievskiV

# import libraries
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation

# ignore warnings
import warnings
warnings.filterwarnings("ignore")

def brownian_motion(N, T, h, seed=42):

IlievskiV

from scipy import stats

num_simulations = 10000  # how many times to repeat
Ns, Ts, hs = 20000, 10.0, 1.0 # discrete steps, continuous steps, veriance
dts = 1.0 * T/N  # total number of time steps

u = 2.  # the difference in time points
t = int(np.floor((np.random.uniform(low=u+0.01, high=1. * T - u)/T) * N)) # random starting point

# initialize the means

IlievskiV

import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation

fig = plt.figure(figsize=(21, 10))
ax = plt.axes(xlim=(0, 1)) 
line, = ax.step([], [], where='mid', color='#0492C2')
    
# formatting options
ax.set_xticks(np.linspace(0,1,11))
ax.set_xlabel('Time', fontsize=30)

IlievskiV

import numpy as np
np.random.seed(1234)

def brownian_motion(N, T, h):
    """
    Simulates a Brownian motion
    :param int N : the number of discrete steps
    :param int T: the number of continuous time steps
    :param float h: the variance of the increments
    """   

IlievskiV

num_simulations = 10000 # num of simulation
N = 100000 # number od steps in each simulation
dt = 1./N # the time step
X_norm = [0] * num_simulations # the normalized random variable

# run the simulations
for i in range(num_simulations):
    X, _ = random_walk(N)
    X_norm[i] = X[N - 1] * np.sqrt(dt)
    

IlievskiV

import matplotlib.pyplot as plt
import matplotlib.animation as animation

fig = plt.figure(figsize=(21, 10))
ax = plt.axes(xlim=(0, N), ylim=(np.min(X) - 0.5, np.max(X) + 0.5)) 
line, = ax.plot([], [], lw=2, color='#0492C2')
ax.set_xticks(np.arange(0, N+1, 50))
ax.set_yticks(np.arange(np.min(X) - 0.5, np.max(X) + 0.5, 0.2))
ax.set_title('2D Random Walk', fontsize=22)
ax.set_xlabel('Steps', fontsize=18)

IlievskiV

import numpy as np
np.random.seed(1234)

def random_walk(N):
    """
    Simulates a discrete random walk
    :param int N : the number of steps to take
    """
    # event space: set of possible increments
    increments = np.array([1, -1])

IlievskiV