import numpy as np
import matplotlib.pyplot as plt
from matplotlib import rc

#graphical parameters
############################################################################### 
plt.style.use('ggplot')
rc('font',**{'family':'serif','serif':['Computer Modern Roman'],
     'size' : '12'})
rc('text', usetex=True)
rc('lines', linewidth=2)
plt.rcParams['axes.facecolor']='w'
###############################################################################

np.random.seed(42)

def brownian_bridge(t,a,b):
    n=np.size(t)-2
    X=np.zeros(n+2,)
    X[0]=a
    X[-1]=b
    tfinal=t[-1]
    Z=np.random.standard_normal(n,)
    #main loop
    for k in range(1,n+1):
        mu = X[k-1] + (b-X[k-1])*(t[k]-t[k-1])/(tfinal-t[k-1]);
        sig2 = (tfinal-t[k])*(t[k]-t[k-1])/(tfinal-t[k-1]);
        X[k] = mu +np.sqrt(sig2)*Z[k-1];
        
    return X
        
# Having defined the function, we are ready to run it

a = np.pi; #initial point
b = np.sqrt(5); #final point
n = 1000;
t = np.linspace(0,1,n);
X = brownian_bridge(t,a,b);
plt.plot(t,X,label='Path');
plt.plot([0,1],[a,b],marker="X",linestyle='None',label='Initial and final points');

plt.xlabel("t");
plt.ylabel("t");
plt.legend()
plt.title('Brownian Bridge')
plt.show()