Up Panel: Multiple simulated paths of the process ${\displaystyle X_{t}}$ which is stopped upon hitting ${\displaystyle -1}$. This shows gambler's ruin behavior, and is not a martingale. Down Panel: Paths of ${\displaystyle X_{t}}$ with an additional stopping criterion: the process is also stopped when it reaches a magnitude of ${\displaystyle k=2.0}$. This no longer suffers from gambler's ruin behavior, and is a martingale.

import numpy as np
import matplotlib.pyplot as plt

dw = 0.05
dt = dw**2
num_samples = 5

def generate_Xt(k=None):
    W = np.array([])
    while True:
        B = np.array([np.random.default_rng().choice([-1, 1]) for _ in range(1000)])
        W = np.append(W, np.cumsum(B) * dw)
        t = np.arange(len(W)) * dt

        if k is not None:
            hitting_index = np.argmax((W >= k)