Stochastic Processes

Random walks, Markov chains, and Poisson processes.

Parameters

Steps Paths p (prob right/up)

2D Walk

Paths Time T Steps N

Geometric BM

μ (drift) σ (volatility) S₀

Rate λ Time T

States N (max) Birth rate λ Death rate μ

States n State names (comma-sep)

Initial distribution π₀ Steps

Start state (0-indexed) Steps

Win prob p Start wealth k Target N

Live Data

E[X_n]—

Var[X_n]—

σ—

Finals: —

E[W(T)]0

Var[W(T)]—

Sim E[W(T)]—

E[N(T)] = λT—

Var[N(T)]—

Simulated N(T)—

Mean inter-arr—

ρ = λ/μ—

π₀—

E[Q]—

See matrix setup above.

Plot displays steady state convergence.

Plot displays state frequencies.

P(win)—

P(ruin)—

See left panel for matrix controls.

Stochastic Processes & Random Walks Handbook

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.