ThmDex – An index of mathematical definitions, results, and conjectures.

Result R5379 on D3657: Real Wiener process

Distribution of the real Wiener process at a given point is gaussian

Formulation 0

Let $W \in \text{Wiener}(\mu, \sigma)$ be a D3657: Real Wiener process.
Let $t \in [0, \infty)$ be an D4767: Unsigned real number.

Then \begin{equation} W_t \overset{d}{=} \text{Gaussian}(\mu t, \sigma \sqrt{t}) \end{equation}

▶▶▶	Comment 0 Note: the second parameter $\sigma \sqrt{t}$ in the notation $\text{Gaussian}(0, \sigma \sqrt{t})$ is the standard deviation so that $\text{Var} W_t = \sigma^2 t$.

Proofs

Proof 0

Let $W \in \text{Wiener}(\mu, \sigma)$ be a D3657: Real Wiener process.
Let $t \in [0, \infty)$ be an D4767: Unsigned real number.

Let $B \in \text{Wiener}(0, 1)$ be a standard Wiener process. By definition \begin{equation} W_t \overset{d}{=} \mu t + \sigma B_t \end{equation} Result R5378: Distribution of the standard real Wiener process at a given point is gaussian shows that \begin{equation} B_t \overset{d}{=} \text{Gaussian}(0, \sqrt{t}) \end{equation} The claim follows. $\square$