Math 2.0

Probability theory and stochastic processes Definition

Wiener process

Builds on Stochastic processNormal distributionIndependence of two random variables

Definition

Let (Ω,ℱ,P)(\Omega, \mathcal{F}, P) be a probability space. A Wiener process, or standard Brownian motion, is a real stochastic process (Wt)t≥0(W_t)_{t \ge 0} such that:

  1. W0=0W_0 = 0 almost surely.
  2. For all 0≤t0<t1<⋯<tn0 \le t_0 < t_1 < \dots < t_n, the increments Wt1−Wt0,…,Wtn−Wtn−1W_{t_1} - W_{t_0}, \dots, W_{t_n} - W_{t_{n-1}} are independent.
  3. For all 0≤s<t0 \le s < t, the increment Wt−WsW_t - W_s has the normal distribution with mean 00 and variance t−st - s.
  4. Almost surely, the path t↦Wtt \mapsto W_t is continuous.

Remarks

Such a process exists; one construction takes the uniform limit of random series ∑kξkϕk(t)\sum_k \xi_k \phi_k(t), with independent standard normal ξk\xi_k and suitable functions ϕk\phi_k such as the Schauder functions. The process Xt=2WtX_t = 2 W_t satisfies clauses 1, 2 and 4 but fails clause 3, since Xt−XsX_t - X_s has variance 4(t−s)4(t - s). Equivalently, WW is a centred Gaussian process with E[WsWt]=min⁡(s,t)E[W_s W_t] = \min(s, t) and continuous paths. A process Wt+μtW_t + \mu t, or σWt\sigma W_t, is called a Brownian motion with drift μ\mu or with variance parameter σ2\sigma^2. A dd-dimensional Wiener process is a vector of dd independent real Wiener processes.

Reading path

To understand this page from the ground up, read these 42 pages in order. The foundations come first.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Independence of two random variables concept
  4. Integers object
  5. Normal distribution concept
  6. Ordered field definition
  7. Proposition concept
  8. Random experiment concept
  9. Rational numbers object
  10. Real numbers object
  11. Set definition
  12. Ring of integers statement
  13. Disjunction definition
  14. Implication definition
  15. Predicate definition
  16. Ordered pair definition
  17. Union definition
  18. Sample space definition
  19. Existential quantifier definition
  20. Universal quantifier definition
  21. The rationals form a field statement
  22. Cartesian product definition
  23. Subset definition
  24. Uniqueness quantifier definition
  25. The rationals form an ordered field statement
  26. Binary relation definition
  27. Universal set definition
  28. Power set definition
  29. Order on the reals definition
  30. Complement definition
  31. Function definition
  32. Interval definition
  33. Sigma-algebra definition
  34. Preimage of a set definition
  35. Event definition
  36. Borel sigma-algebra definition
  37. Probability axioms concept
  38. Probability measure definition
  39. Probability space definition
  40. Random variable definition
  41. Stochastic process concept
  42. Wiener process definition

Where this leads

8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Wiener_process
About this page
Tag552AWX3QXN
URIm2://552AWX3QXN
Kinddefinition
WritingW0 stub
MSC60J65, 60G15
DefinesWiener process
UsesStochastic process, Normal distribution, Independence of two random variables
Changed2 November 2026

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