Stochastic Processes and Simulation Calculators
This page presents tools that summarise random movement, state transitions and simulated outcomes, providing numerical results for systems influenced by uncertainty.
-
Geometric Brownian Motion (GBM) Calculator
A growth process with random movement produces outcomes that vary around a long-term trend shaped by both average change and uncertainty.
Example use: estimating how a repeatedly adjusted household reading might vary when each change includes a small random shift.
Inputs: initial value, average rate of change, volatility level, total time, number of steps, number of simulated paths
Outputs: mean final value, probability of ending above the starting point, sample median, average peak-to-trough decline, theoretical median, realised average rate of change, standard deviation, highest sample value, skewness, lowest sample value, excess kurtosis, interquartile range
Visual: simulated paths over time, a distribution of final values, and a comparison between simulated and expected long-term behaviour
-
Hidden Markov Model (HMM) Calculator
A system that moves between unobserved states produces visible outcomes whose likelihood depends on both the current state and the chance of switching to another.
Example use: interpreting a sequence of daily temperature descriptions to infer the most likely underlying weather pattern.
Inputs: transition probabilities, emission probabilities, starting state, state names, observation symbols, observation sequence indices
Outputs: sequence likelihood, log likelihood, most likely state path probability, most likely state path
Visual: a decoded state sequence and a colour-based display of transition strengths
-
Kalman Filter Calculator
A step-by-step updating method refines an estimate by combining uncertain measurements with a predicted value.
Example use: smoothing a series of noisy household thermometer readings to obtain a clearer temperature estimate.
Inputs: data input type, sample data, raw data, base value, number of steps, measurement noise, process noise
Outputs: final filtered estimate, root mean square error, mean absolute error, settling step, average gain, signal-to-noise improvement, variance reduction
Visual: a comparison of measured values and the filtered estimate across the steps
-
Markov Chain Simulation Calculator
A sequence of outcomes determined only by the current state produces a pattern of transitions that settles into a long-run distribution.
Example use: modelling how a person might rotate through a set of preferred evening activities based on their current choice.
Inputs: transition matrix, initial state vector, state names, number of simulation steps
Outputs: state after each step, final probability distribution, steady-state distribution, return time, status, step count, calculation process
Visual: a line display of state probabilities over time and a colour-based view of transition strengths
-
Monte Carlo Simulation Calculator
A repeated random sampling process produces a spread of possible outcomes based on chosen step changes and uncertainty levels.
Example use: estimating how a series of daily step counts might vary when each day includes a random change.
Inputs: number of simulations, steps per simulation, initial value, average step change, step standard deviation, probability distribution type, degrees of freedom
Outputs: mean final value, standard deviation, median, minimum, maximum, fifth percentile, twenty-fifth percentile, seventy-fifth percentile, ninety-fifth percentile, skewness, excess kurtosis, probability of ending above the starting value
Visual: simulated paths over time and a distribution of final outcomes
-
Ornstein-Uhlenbeck (OU) Process Calculator
A mean-reverting process moves randomly while being pulled toward a long-term average at a rate determined by its reversion strength.
Example use: modelling how a room's indoor humidity might drift but gradually return toward a typical level.
Inputs: initial value, long-term average, reversion speed, volatility, total time, number of steps, number of simulations
Outputs: average final value, median final value, standard deviation, mean absolute deviation, interquartile range, skewness, excess kurtosis, half-life of reversion, stationary variance, persistence, average number of crossings of the long-term average, step-based reversion speed, theoretical variance, variance fit measure, initialisation steps, time increment, persistence calculation, half-life calculation, simulation results, final value analysis, empirical variance comparison
Visual: simulated paths with an average path line, a comparison of simulated and theoretical distributions, and a comparison of empirical and theoretical variance over time
-
Standard Brownian Motion
A continuous random movement with no overall drift produces paths that wander unpredictably while maintaining a characteristic spread.
Example use: representing how a person's position might change when taking steps in random directions during a walk.
Inputs: initial value, total time, number of steps, number of simulations
Outputs: average final value, maximum final value, minimum final value, standard deviation, skewness, kurtosis, mean absolute deviation, median final value, interquartile range, average running maximum, average time spent above the starting point, average quadratic variation, mean square displacement, variance trajectory fit
Visual: simulated paths, a distribution of final values, a variance-over-time display, and a two-dimensional random walk illustration
Stochastic Processes and Simulation FAQs
Stochastic processes describe systems that evolve with randomness, presenting values that change unpredictably across time steps.
Simulation calculators generate sequences or distributions of outcomes, showing how random variation influences overall system behaviour.
A Markov chain represents transitions between states where each move depends solely on the current state rather than earlier history.
Monte Carlo simulation repeats random sampling many times to approximate possible outcomes, producing a distribution rather than a single estimate.
Mean reversion describes movement toward a long-term central value, with deviations gradually pulled back toward the average over time.