WienerProcess

Standard Brownian motion with drift $\mu$ and volatility $\sigma$, starting at $0$.

WienerProcess()standard Wiener process, $\mu = 0$, $\sigma = 1$.

WienerProcess(mu, sigma)an inert process object — carries its two parameters, unevaluated. SliceDistribution and RandomFunction both read it.

Typed signatures

Type	Package	Applies
(real?, real<0..>?) -> expression<WienerProcess>	statistics	always

Domain: Statistics

Details

• Wolfram writes a process's value at time $t$ as proc[t], a call, which reads as SliceDistribution(proc, t).
• $SliceDistribution(WienerProcess(\mu,\sigma), t) = NormalDistribution(\mu t, \sigma\sqrt{t})$ — exact, since a Wiener process's marginal at any fixed $t$ is Gaussian by definition.
• RandomFunction(WienerProcess(mu, sigma), {tmin, tmax, dt}) draws one seeded sample path by Euler–Maruyama over the requested grid, starting from $X(tmin) = 0$ — exact in distribution for this process, since its increments are themselves exactly Gaussian.

Examples

Basic #

$Mean(WienerProcess(\mu,\sigma)[t]) = \mu t$

live ↗

In$\mathrm{mean}(\mathrm{SliceDistribution}(\mathrm{WienerProcess}(2, 3), 5))$Out$10$

$Variance(WienerProcess(\mu,\sigma)[t]) = \sigma^2 t$

live ↗

In$\mathrm{Variance}(\mathrm{SliceDistribution}(\mathrm{WienerProcess}(0, 3), 4))$Out$36$

Scope #

Standard Wiener process — $NormalDistribution(0, \sqrt{4}) = NormalDistribution(0, 2)$

live ↗

In$\mathrm{SliceDistribution}(\mathrm{WienerProcess}(), 4)$Out$\operatorname{NormalDistribution}(0, 2)$

live ↗

In$\operatorname{CDF}(\mathrm{WienerProcess}()\lhd2, x)$Out$\frac{1}{2}(\operatorname{erf}(\frac{x}{2})+1)$

live ↗

In$\operatorname{PDF}(\mathrm{WienerProcess}()\lhd2, x)$Out$\frac{\exp(-\frac{x^2}{4})}{2\sqrt{π}}$

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In$\operatorname{PDF}(\mathrm{WienerProcess}(1, 1)\lhd t, x)$Out$\frac{\sqrt{2}\exp(-\frac{(x-t)^2}{2t})}{2\sqrt{π}\sqrt{t}}$

About

Examples	6
Checked against	Wolfram Language 3 agree

See also: PoissonProcess, SliceDistribution, RandomFunction, NormalDistribution