STOCHASTIC PROCESSES 04

Random Trajectories

A trajectory is one realised evolution of a stochastic process through time.

One realisation

A stochastic process specifies many possible outcomes at many times. Once one random evolution occurs, we obtain a sample path

\[t\mapsto X_t(\omega).\]

Here \(\omega\) represents one outcome of the underlying random experiment.

1. Different runs give different paths

If the same stochastic model is simulated repeatedly from the same initial state, the trajectories can differ because different random events occur.

2. A trajectory is not the distribution

One path shows one possible history. It does not by itself tell us the probabilities of all possible histories.

\[\boxed{\text{one trajectory}\ne\text{the stochastic model}}\]

3. Ensemble

A collection of independent trajectories generated from the same model is often called an ensemble. It can be used to estimate probabilities, expectations and variances.

4. Mean trajectory

At time \(t\), the expected value

\[E[X_t]\]

is an average across the distribution of possible values at that time. It need not coincide with any individual trajectory.

5. Jump and continuous paths

A CTMC has piecewise-constant paths with random jumps. Brownian motion has continuous but highly irregular paths.

6. Simulation

Simulation generates sample paths. Repeating simulations many times provides numerical information about the distribution of the process.

\[\boxed{\text{model}\rightarrow\text{random numbers}\rightarrow\text{trajectory}\rightarrow\text{many runs}\rightarrow\text{distributional estimates}}\]