Stochastic Processes

Mathematical Principles & Theorems

Simulates continuous and discrete stochastic processes: (1) Wiener Process / Brownian Motion \(W(t)\): continuous-time process with independent Gaussian increments \(W(t+s)-W(t) \sim \mathcal{N}(0, s)\); (2) Geometric Brownian Motion \(dS_t = \mu S_t dt + \sigma S_t dW_t\) modeling asset prices; (3) Discrete Random Walk \(S_n = \sum_{i=1}^n X_i\) on \(\mathbb{Z}^d\); (4) Discrete-Time Markov Chains with transition probability matrix \(P = [P_{ij}]\), stationary distribution \(\boldsymbol{\pi} P = \boldsymbol{\pi}\), and absorbing state absorption probabilities.

Operating Instructions

• Select stochastic model: Brownian Motion, Geometric Brownian Motion, Random Walk, or Markov Chain.
• Configure drift \(\mu\), volatility \(\sigma\), step count, and Monte Carlo ensemble size via numeric inputs.
• Click Run Simulation to generate stochastic path trajectories in real time.
• Inspect sample path ensembles, mean trajectory envelopes, and terminal variance distributions.