4. Stochastic processes

Definition of stochastic process.

Examples.

• White noise, \(\xi(t)\), is a zero-mean process with no correlation between its values at different times

  \[\mathbb{E}\left[ \xi(t) \, \xi(s) \right] = \delta(t-s)\]

• Wiener process (Brownian motion), \(W(t)\)

  \[\begin{split}\begin{aligned} & W(0) = 0 \\ & W(t) \text{ has independent increments} \\ & W(t) - W(s) \sim N(0, t-s) \text{ for $t > s$} \\ & W(t) \text{ are continuous but nowhere differentiable} \end{aligned}\end{split}\]

  Informal relation between Wiener process and white noise signal

  \[\begin{split}\begin{aligned} W(t) - W(s) & = \int_{s}^{t} \xi(\tau) \, d \tau \\ {}^{''} \dfrac{d W(t)}{d t} & = \xi(t) {}^{``} \end{aligned}\end{split}\]

  where the derivative relation doesn’t hold in the classical sense, as \(W(t)\) is nowhere differentiable

• time-discrete Markov processes

Applications

• LTI

• Stochastic differential equations…

  \[d X(t) = \mu(t) \, dt + \sigma(t) \, dW(t)\]

Assumptions.

• Stationariety

• Ergodicity

\[k_{xy}(\tau) := \mathbb{E}[x(t) y(t-\tau)] = \lim_{T\rightarrow +\infty} \left\{ \frac{1}{2T} \int_{t = -T}^{T} x(t) \, y(t-\tau) \, d t \right\}\]