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Brownian Motion

Nov 14, 20242 min read

ProbabilityTheoryStochasticProcesses A Brownian Motion is a Stochastic Process such that

  1. is an Adapted Process
  2. (starts from 0)
  3. and for (increments are Gaussian)
  4. for (independent increments)
  5. is continuous for a.a. (continuous sample paths)

In addition:

  • A Brownian Motion started from is defined such that is a Brownian Motion
  • A -dimensional Brownian Motion is such that are Brownian Motions for
  • If , then 3. can be written .

A Brownian motion has the following properties:

Misplaced & \limsup_{t \to \infty} W_t &= \infty \\ \liminf_{t \to \infty} W_t &= -\infty \\ W_t &\sim \sqrt{t}\\ W_t &= o(\frac1t) \end{aligned}$$ - **Approximately Hölder sample paths**, for $\delta \in (0, \frac12)$: $$|B_t - B_s| \leq C|t-s|^{\frac12 - \delta}$$ - **Positive and negative on any time-interval**: - **Reflection principle**:

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