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.

```python 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: