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The Fundamental Properties of Wiener Processes

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The Formal Theorem

A stochastic process W={Wt:t≥0} W = \{W_t : t \ge 0\} is defined as a standard Wiener process if it satisfies the following axioms: (i) W0=0 W_0 = 0 almost surely. (ii) The increments Wt−Ws W_t - W_s are independent for all 0≤s<t 0 \le s < t . (iii) For any 0≤s<t 0 \le s < t , the increment follows a normal distribution:
Wt−Ws∼N(0,t−s) W_t - W_s \sim \mathcal{N}(0, t-s)
(iv) The sample paths t↦Wt t \mapsto W_t are continuous almost surely.

Analytical Intuition.

Imagine a microscopic particle suspended in a fluid, buffeted by a relentless, invisible storm of molecular collisions. This is Brownian motion, the physical embodiment of the Wiener process. At any time t t , the particle's position is an aggregation of infinite, infinitesimal independent shocks. The beauty lies in the scaling: because the variance of the displacement grows linearly with time—specifically Var(Wt)=t Var(W_t) = t —the process exhibits self-similarity. If you zoom into a segment of the path, it looks statistically identical to the whole. Despite the path being everywhere continuous, it is nowhere differentiable; it is a jagged, fractal trajectory where the velocity is undefined at every point. We are observing the limit of a random walk where the step size vanishes and frequency diverges, resulting in a process that is essentially Gaussian noise integrated over time. It is the fundamental building block of stochastic calculus, providing the 'rough' landscape upon which the Ito integral is constructed, allowing us to perform analysis in systems driven by pure, unpredictable volatility.
CAUTION

Institutional Warning.

Students frequently conflate the 'nowhere differentiability' of Wt W_t with 'discontinuity.' While the path is indeed continuous, its variation is infinite; it has no tangent. It is a continuous function that is 'fractal' in its lack of smoothness.

Academic Inquiries.

01

Why is the variance t t and not t \sqrt{t} ?

The standard deviation is t \sqrt{t} , but the variance E[Wt2] E[W_t^2] of the increment Wt−W0 W_t - W_0 is t t because the process is defined by the accumulation of independent shocks, making the variance additive.

02

Does the Wiener process have finite variation?

No. The quadratic variation of a Wiener process over [0,T] [0, T] is T T , and its first variation is infinite, which is why standard Riemann-Stieltjes integration fails.

Standardized References.

  • Definitive Institutional SourceØksendal, B., Stochastic Differential Equations: An Introduction with Applications.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). The Fundamental Properties of Wiener Processes: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/advanced-stochastic-processes/the-fundamental-properties-of-wiener-processes

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